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This book contains 40 delightful paradoxes. Here is a small sampling.LOGIC: Is it ever right to ask the question: "May I disturb you?" The very act of asking will disturb the person. And yet, I simply can't know if it's correct to ask the question unless I actually ask the question!PROBABILITY: In 2007, the college football team USC was ranked as 7th in the Harris poll, 6th in the USA Today poll, and 6th in the computer rankings. And yet, when the three polls were averaged, USC ended up as being ranked as the 5th best team overall. How is that possible?GAME THEORY: You play game A that is a losing bet. You also play game B that is a losing bet. Yet when you play games A and B alternately that is a winning bet. How can two losing games combine to make a winning game?
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40 Paradoxes in Logic, Probability, and Game Theory

Presh Talwalkar

Copyright Presh Talwalkar 2013

About The Author

Presh Talwalkar studied Economics and Mathematics at Stanford University. His site Mind Your Decisions has blog posts and original videos about math that have been viewed millions of times.

Books By Presh Talwalkar

The Joy of Game Theory: An Introduction to Strategic Thinking . Game Theory is the study of interactive decision-making, situations where the choice of each person influences the outcome for the group. This book is an innovative approach to game theory that explains strategic games and shows how you can make better decisions by changing the game.

Math Puzzles Volume 1: Classic Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory . This book contains 70 interesting brain-teasers.

Math Puzzles Volume 2: More Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory . This is a follow-up puzzle book with more delightful problems.

Math Puzzles Volume 3: Even More Riddles And Brain Teasers In Geometry, Logic, Number Theory, And Probability . This is the third in the series with 70 more problems.

But I only got the soup! This fun book discusses the mathematics of splitting the bill fairly.

40 Paradoxes in Logic, Probability, and Game Theory . Is it ever logically correct to ask “May I disturb you?” How can a football team be ranked 6th or worse in several polls, but end up as 5th overall when the polls are averaged? These are a few of the thought-provoking paradoxes covered in the book.

Multiply By Lines . It is possible to multiply large numbers simply by drawing lines and counting intersections. Some people call it “how the Japanese multiply” or “Chinese stick multiplication.” This book is a reference guide for how to do the method and why it works.

The Best Mental Math Tricks . Can you multiply 97 by 96 in your head? Or can you figure out the day of the week when you are given a date? This book is a collection of methods that wi; ll help you solve math problems in your head and make you look like a genius.





Table of Contents

Introduction

LOGICAL PARADOXES

1) Knowing Without Knowing

2) The Case of the Missing Dollar

3) Lowering Both Averages

4) What Exactly is a Pile of Sand?

5) The Missing Square

6) “Proofs” that 1 = 2

7) The Least Interesting Number

8) Selected Self-Referential Paradoxes

9) The Debt that Must be Paid?

10) The Paradox of the Question

11) Russell's Set of All Sets

12) A Paradox of Motion

13) How Can a Part be Equal to the Whole?

14) Infinite Series Paradox

PROBABILITY PARADOXES

15) Birthday Paradox

16) False Positive Paradox

17) Simpson's Paradox

18) Bertrand's Box Paradox

19) Boy or Girl Paradox

20) The Monty Hall Problem

21) The Necktie Paradox

22) The Friendship Paradox

23) St. Petersburg Paradox

24) Wild Card Poker Paradox

25) Gambler's Fallacy

26) Surprise Quiz Paradox

27) Two Envelopes

28) Siegel's Paradox of Exchange Rates

29) Allais Paradox

30) Ellsberg Paradox

31) Random Chord

GAME THEORY PARADOXES

32) You Can't Choose Wrong Paradox

33) Parrondo's Paradox

34) Centipede Game

35) Hypergame Paradox

36) The Wallet Paradox

37) Condorcet Paradox

38) Apportionment Paradox

39) Forward Induction

40) Braess Paradox

Conclusion





Introduction

The word “paradox” comes from the Greek roots para (“against”) and doxa (“belief”). As used in this book, a paradox is a conclusion that is contradictory, counter-intuitive, or surprising.

This book is a collection of mostly logical and mathematical paradoxes that range from easy to understand and fun to resolve to bewilderingly hard to grasp and impossible to resolve.

The paradoxes are organized into three sections:

(1) Logical paradoxes where reasoning leads to contradictory results.

(2) Probability paradoxes that demonstrate the counter-intuitive nature of randomness.

(3) Game theory paradoxes in which “irrational” strategies are surprisingly the “rational” option.

I have provided notes for each section that include citations and sources. It is not a traditional bibliography with consistent formatting, but rather a guide with enough information to be able to track down the original sources.

I love getting feedback. If you have a comment or suggestion, please email me presh@mindyourdecisions.com





LOGICAL PARADOXES

This section is an assortment of mostly logical paradoxes, covering four main topics.

The opening paradoxes are about the difficulty in following logic precisely (like the Paradox of Knowing without Knowing ). The next few are concepts that seem impossible to pin down with numbers (What is the Least Interesting Number? ). Then there are a few Self-Referential Paradoxes (do you question the rule to “question everything”?). Finally, this section concludes with a few mathematical paradoxes involving infinity.





1)Knowing Without Knowing

What distinguishes art from pornography? The difference is hard to define because even great artists have depicted nudity (Michelangelo’s David ) and used obscene language (Shakespeare1 “peppered his plays with profanity”).

Furthermore, the question is important because art is protected free speech while pornography is not and can be censored.

The question was brought to the U.S. Supreme Court in the1964 case Jacobellis v. Ohio . The manager of an arts theater, Nico Jacobellis, was fined by the state of Ohio for showing a French film considered obscene. Jacobellis contended the film was art, and as free speech, the state had no right to restrict it. Ultimately Jacobellis won the case, but there was a larger issue at hand. Would the Supreme Court explain the difference between art and pornography?

In short, no. The court opinions were divided and skirted the topic. Famously, Justice Potter Stewart dodged the issue by writing2 , “I shall not today attempt further to define [pornography] … and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that.”

The ruling ended the case at hand, but it raised a logical dilemma. How was someone supposed to avoid pornography? Well, in order to view only art, one has to be able to identify pornography. But in order to identify pornography, one has to view it. The rule “I know it when I see it” does not settle the matters, but instead translates it into an unsolvable chicken and egg paradox.

The identification problem is not just limited to art and pornography. The paradox is that often one needs to know the answer to act, but it is only possible to learn the answer by acting. In other words, one needs to have the impossible talent of “knowing without knowing.”

Here are six more examples of the paradox.

(1) The Fifth Amendment of the United States Constitution provides that people have the right not to self-incriminate themselves in court. But how can a judge know in advance if the testimony would be self-incriminatory? The only way to know is by finding out what the circumstances are. But often it is the circumstances that the person wishes to avoid divulging. Judge Learned Hand noted the paradox3 : "Logically, indeed, [the witness] is boxed in a paradox, for he must prove the criminatory character of what it is his privilege to protect because it is criminatory."

(2) When is it correct to ask the question “May I disturb you?” The question is troublesome because the very act of asking it requires disturbing the listener. The question should really only be asked to someone who is okay being disturbed. But there is no way of knowing that in advance! The only way to find out if someone can be disturbed is by disturbing them, which essentially ruins the polite intention of the question.

(3) Companies advertise “You will love it” to get people to buy products, try new food items, and see movies. And yet, the company cannot necessarily know how people will react. The only way someone can actually say they love it is if they try it. And yet, in order for someone to want to try something new, they have to reasonably think they will like it, which is why the company needs to tell them “You will love it.”

(4) In the TV show The Simpsons , Homer offers to tell his wife Marge a secret, but only if she promises in advance that she will not get mad. How can Marge know if she will get mad? She can only genuinely react to the news once she hears it; but she has to commit in advance in order to hear the news. Marge humorously replies that she will get mad, because she always does get mad when she has to make a promise that she will not. The idea of promising to react a certain way is a common joke in many TV shows and films4 .

(5) Imagine a picture of your private parts appears on the internet. You know the picture is a fake because it is anatomically wrong. But how can you prove it? You could release a genuine picture to remove any doubt, but that would obviously be counter-productive since exposing yourself publicly is precisely what you wish to avoid!

(6) The website Scribd is a document-sharing website. The website ran into trouble because users were uploading copyrighted material, such as leaking new Harry Potter books. In 2009 a lawsuit5 alleged Scribd was encouraging copyright infringement. Scribd replied that it did its best to filter out copyrighted material, and in fact, it was checking documents against an internal database. The funny part was this: the lawsuit alleged that the database was an act of copyright infringement too! The plaintiff's claim was that Scribd was supposed to filter out copyrighted material, but that it had to do so without a database to check against. How was Scribd supposed to know what is copyrighted without having copies of the copyrighted material?

The heart of the paradox is that knowledge is limited and you often cannot know the correct answer in advance. You have to do something and then later find out if it was justified.

In reality, people have to operate with limited information. In the examples above, here are some of the possible remedies.

(1) The Fifth Amendment privilege is often taken into context. The person might be able to provide non-relevant details to establish the testimony would be self-incriminatory, or the prosecutor might have reason to believe the person is lying and should be forced to testify on certain details.

(2) The question “May I disturb you?” is often not taken literally. The question is often a form of etiquette to mean, “I know you look busy, so let me check; are you really?” Someone who cannot be interrupted may simply remain silent and continue working as a signal that they are too busy to be disturbed. Someone who can be interrupted will feel obliged by the courtesy. Therefore, asking the question “May I disturb you?” is often better than simply approaching someone with no warning.

(3) Companies create previews of new products so people can “know” they will enjoy something before they have actually bought the product. There are a variety of preview methods. A technology company might do a live preview of their product at a conference. A company selling a subscription good might offer a free trial-period. A food company can distribute free samples. A final common example is known to any movie-goer. Movie trailers are created so a viewer can get a sense of a film's plot without having to see the entire movie. All of these methods entice and persuade people so they will want a product before having the experience of buying and using it .

(4) Like the question “May I disturb you,” the request, “Promise me you will not...” is also not taken at face value. The requester wishes to signal the information is somehow embarrassing, hurtful, or confidential. The requester knows the person answering might not live up to the promise, but at least the stipulation was made explicit. Someone who breaks a promise one time might not be given a chance another time.

(5) People try to prove graphic pictures are fake by destroying the credibility of the person who leaked the picture. They also try to demonstrate their own morality and explain why they would never do such a thing.

(6) The lawsuit against Scribd for using a copyright filtering system was dropped6 . Publishers realized that Scribd does not have a magic way of filtering out copyrighted material.

In conclusion, while the paradox of “knowing without knowing” cannot be completely resolved, there are some practical workarounds that at least make the circumstances less difficult, and life a bit easier.

Notes

1. Angier, Natalie. “Almost Before We Spoke, We Swore.” The New York Times. 20 Sept. 2005. Web http://nyti.ms/XebEa1

2. Jacobellis v. Ohio , 378 U.S. 184 (1964). http://supreme.justia.com/cases/federal/us/378/184/case.html

3. Justice Learned Hand quoted from Suber, Peter. The Paradox of Self-Amendment. Available online. http://legacy.earlham.edu/~peters/writing/psa/sec20.htm

4. The Simpsons , “Lisa the Greek.” Quoted at TV Tropes . http://tvtropes.org/pmwiki/pmwiki.php/Main/PromiseMeYouWontX

5. Johnson, Bobbie. “Book sharing site Scribd rejects claims of copyright infringement.” Gaurdian.co.uk . 21 Sept. 2009. http://www.guardian.co.uk/technology/2009/sep/21/scribd-lawsuit

6. Kravets, David. “Lawsuit Dropped; Claimed That Copyright-Filtering Violates Copyright.” Wired. 19 Jul 2010. http://www.wired.com/threatlevel/2010/07/copyrightfiltering-scribd/





2)The Case of the Missing Dollar

It was midnight and my consulting team was still in the office. We were working on an accounting case and we needed to audit our results using two equivalent derivations. But time and again when we crunched the numbers, we kept finding discrepancies. My boss joked, “It's like the story with the missing dollar.” We had all heard this puzzle before, and yet in our sleep-deprived and mentally exhausted states, the riddle left us confused.

This is a classic accounting problem. Can you figure it out?

The story

Three gentlemen stay in a hotel room. The bill is $30 so each of them pays $10 for the room. The next day the hotel manager realizes he has overcharged them. The bill should actually be $25. The hotel manager gives $5 to the bellboy and tells him to return the money.

The bellboy thinks about how $5 cannot be split evenly in three ways. Since the gentlemen don't know about the correction, he decides he can pocket $2 and return $1 to each person.

Now each of the three gentlemen has paid $9 for his room, so that makes $27. In addition, the bellboy has pocketed $2. So here's the puzzle: the $27 they paid and the $2 the bellboy pocketed only adds up to $29. But the room originally cost $30. Where did the missing dollar go?

Try solving it before reading the answer.

The answer

Here's the reason: the story is a tale of misdirection because, of course, there is no missing dollar.

One method is to account from the original bill. Starting with the $30 paid, $3 made it to the gentlemen, lowering the total to $27. Additionally, the bellboy pocketed another $2, bringing the net balance to $25. This exactly matches the amount on the revised bill.

The alternate method is account from the revised bill. The logic is exactly the same but the steps are reversed. The final bill was $25, from which $2 was pocketed by the bellboy, and also $3 were refunded to the gentlemen. Adding the three brings the total to $30, which was the amount of the original bill.

The apparent mystery is a result of confusing the two methods. The accounting in the story takes the $27 paid and then incorrectly adds the $2 the bellboy pocketed to arrive at $29. This is just wrong. Either one adds the $3 refund to the $27 to arrive at $30, or one subtracts the $2 pocketed from $27 to arrive at $25.

The story demonstrates why it is important to carefully account for cash flows—it is easy to get confused!





3)Lowering Both Averages

Consider two teams A and B where the average weight of the players on team A is 200 and on team B is 220 pounds.

What might happen to the average weight of each team if a player is transferred from team A to team B?

Intuition suggests that if the average weight of one team goes down, then the average weight of the other team should go up, or vice versa. After all, if a weight is transferred from one side of a weighing balance to another, one side would fall and the other would rise.

But that does not always happen when considering the average weight!

In fact, there are paradoxical switches that would make the average weight of both teams decrease (or equivalently make the average weight of both teams increase).

How is that possible?

Both averages decrease

Here is a mathematical example that illustrates the paradoxical change.

Suppose the players on each team A have weights {200, 200, 190, 210}, which is an average weight of 200 pounds.

Suppose players on team B have weights {220, 220, 190, 250}, which is an average weight of 220 pounds.

Now transfer the player who weighs 210 pounds from team A to B. What will the new averages be?

Team A now has players with weights {200, 200, 190}, which is an average weight of 196.67—clearly less than the average of 200 pounds they started with.

Team B, on the other hand, has players with weights {220, 220, 190, 250, 210}. The average weight of the team is now 218 pounds, which is also smaller than its starting average of 220.

So the average weight on both teams actually decreases!

Note that one can reverse the transaction to find an example of how the average weight on both teams can increase. (That is, transfer the player who weighs 210 back to team A. Now by reversing the calculations above one can see both teams have increased their average weights).

How is this possible?

The reason this is happening is pretty straightforward. The important factor is that the two averages were not the same to begin with.

When the 210 pound player was removed from team A, the team lost its heaviest player and so naturally that would lower the average weight.

At the same time, introducing the 210 pound player on team B was adding a player whose weight was less than the starting average of 220. Obviously this has the effect of lowering the average weight.

So the trick was this: the player that moved teams had a weight in between the starting average values of the two teams . That had the effect of moving both averages in the same direction—it decreased the average for the team that lost its above-average value, and it lowered the average for the team that got a sub-average value.

What this paradox teaches us

The counter-intuitive result that both averages could increase (or decrease) is known as the Will Rogers phenomenon. Will Rogers was an author and a humorist. During the 1930s Depression, some workers from Oklahoma moved to California in search of jobs. Rogers is quoted as saying1 , “When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states.” The statement is a joke to indicate that while the “Okies” (people from Oklahoma) were a group of below-average intelligence people in Oklahoma, they were still smart enough to be classified as above-average in California. Will Rogers was born in Oklahoma.

The Will Rogers phenomenon is an important statistical concept in medicine. For instance, consider the above example re-framed with team A and team B being two different states. If a group of heavy people from state A migrated to an even heavier state B, statistics might bear out that the average weight in both states was decreasing. It could be easy to conclude that both states were getting “healthier.” In fact, the Will Rogers phenomenon demonstrates the improvement might be illusory; the change could just be a consequence of regrouping the data.

Notes

1. The “Okies” quote is attributed to Will Rogers. It appears in a 1985 medical paper by Dr. Alvan Feinstein, et. al. in the New England Journal of Medicine, “The Will Rogers Phenomenon — Stage Migration and New Diagnostic Techniques as a Source of Misleading Statistics for Survival in Cancer.” Feinstein proposed the term to point out potentially illusory progress in the treatment of cancer patients. The original paper: http://www.nejm.org/doi/full/10.1056/NEJM198506203122504 . I learned about the original source from a 2009 paper by MP Sormani, Ph.D., The Will Rogers phenomenon: the effect of different diagnostic criteria . http://www.ncbi.nlm.nih.gov/pubmed/20106348





4)What Exactly is a Pile of Sand?

A grain of sand is just a grain of sand. And just a few grains of sand are an insignificant collection of sand as well. But if enough grains of sand were present together, then eventually the group will be considered a pile of sand. The question is, at what point does the collection become a pile? No individual grain seems to create the mound, and yet eventually a pile of sand is created. Which grain is the pivotal one?

The Greeks called this the Paradox of Sorites (roughly translated as “paradox of the heap”). The heart of the paradox is that each individual grain of sand has a negligible contribution, but enough grains together will make a pile.

The paradox is a matter of vagueness. What exactly is a pile of sand? One could, for instance, do a survey of people to determine what amount of sand constitutes a pile. From there, one could quantify the subjective opinions and count how many grains are necessary for a pile. But then again, it seems hard to determine a specific standard. If the study concluded that 10,000 grains were needed to make a pile, it would be very likely a collection with one grain fewer, 9,999 grains, would also be considered a pile, so the paradox remains. It seems impossible to demarcate a lower bound because a pile of n grains and a pile of n – 1 grains are virtually indistinguishable.

The problem of the heap applies to any situation in which an individual component makes a seemingly negligible contribution but a large enough collection is significant. Here are 8 similar problems of vagueness.

(1) A single voter almost never sways a national election, but enough voters could change the outcome.

(2) Eating an extra calorie will not ruin a diet, but a collection of snacks can cause weight gain.

(3) A single molecule of air is insignificant, but enough of them can make a tornado or hurricane.

(4) A pinch of spice might not affect the flavor of a dish, but enough spice will .

(5) An extra credit point often has an insignificant impact on a class grade, but earning sufficient marks is the difference between an A and B grade.

(6) A butcher will happily offer a slice or two of meat for a sample, but if he keeps giving slices away he will go bankrupt.

(7) Losing a single strand of hair means nothing, but losing enough hair makes one go bald.

(8) Having an extra sip of beer won't affect one's ability to drive, but at some point imbibing alcohol does impair coordination, making driving unsafe.

Resolving the paradox

In the above examples, each individual action has a negligible impact, but a collection of actions is a slippery slope to a stark outcome.

One possible remedy is to create a demarcation point, even if that point is somewhat arbitrary. For instance, here are some standards commonly used in the eight examples.

(1) A bloc of 10,000 voters in a swing state might be considered influential.

(2) A diet might demand a specific calorie limit to counter the temptation of small snacks.

(3) Meteorologists consider certain wind speeds (say 20 mph and higher) to be dangerous.

(4) A certain amount of spice, like 1/8 of a teaspoon, could be considered a pinch.

(5) Teachers set specific percentage breakdowns for grades.

(6) Butchers will get fed up if a patron asks for more than 5 samples (or a certain number) without buying anything.

(7) People ask around if their hair is “thinning” and realize they are going bald .

(8) The government sets a specific alcohol tolerance level for drivers—or perhaps they might even say there is zero tolerance for any amount.

What this paradox teaches us

Specific standards are easy to mock. After all, why does a diet specify exactly 1,600 calories, or why does the U.S. government think that a driver is safe at a blood alcohol content of 0.079, just below the legal limit of 0.08, but suddenly the driver is dangerous at 0.081? And why does getting 90 percent mean an A grade but getting 89 percent mean a B grade? Did scoring just one percentage point less mean the student is not up to par?

It is easy to ridicule specific standards. But this paradox demonstrates exactly why these standards might be needed: without a clear standard, there can be no objective way of distinguishing one outcome from another.

So we are left to choose between an objective standard or remain in a vague situation where everything can be argued. It appears there is a good reason most organizations opt for the clear standards, even if they might suffer from being arbitrary.





5)The Missing Square

This is a visual paradox. Below are two figures decomposed into smaller shapes.

Why is there a “missing” square in the second figure?



A mathematical excursion

Remember the Fibonacci numbers? The sequence starts out with the numbers 1 and 1, and then each subsequent number is the sum of the previous two numbers.

The next numbers are 2 = 1 + 1, then 3 = 2 + 1, then 5 = 2 + 3, and so on. The sequence is 1, 1, 2, 3, 5, 8, 13, and so on.

There are many interesting properties about the Fibonacci sequence. The one that is relevant to this paradox is a special relation between the numbers.

First a bit of notation. Write F 1 for the first Fibonacci number, F 2 for the second, and generally F n for the n th term. There is a special equation known as Cassini's formula that relates three sequential terms.

The formula is: F n+1 F n-1 – (F n )2 = (-1)n , for n > 0

The formula states that the square of a Fibonacci number is equal to the product of the preceding and succeeding terms, plus or minus a factor of 1.

Why does this formula matter for this problem?

Take a look at the shapes in the figure. All of the shapes have length or widths of 2, 5, 8, or 13, which are Fibonacci numbers. The trick is based on Cassini's formula that (13)(5) – (8)(8) = 1

That extra “1” is essentially where the missing square comes from.

Here is a bit more of the explanation. The top triangle is a true triangle with a height of 5 and a length of 13, so its area is half of (13)(5).

The bottom triangle is not really a triangle. It is actually close in area to half of an 8 by 8 square, with one square removed. The problem utilizes the fact that (8)(8) is one less than (13)(5), which is what makes the optical illusion convincing and difficult to detect.

In the following figure, the two shapes are juxtaposed. The upper left shape looks like a right triangle. The bottom shape has the missing square.

In between the two figures there is a small sliver of area (it's very small so it's hard to show graphically). This area which is shaved is the reason for the “missing” square.



In conclusion, the missing square is not really missing at all. It is just the circumstances create an illusion of a missing quantity.

And a concluding bit of trivia: the Indian mathematician Pingala came upon the same sequence of numbers 1,200 years before Fibonacci in the study of Sanskrit poetry.





6)“Proofs” that 1 = 2

These are not true paradoxes because the conclusions are clearly false. And yet these types of problems are interesting because they take a little bit of thought to figure out the mistake.

Often, such a problem is presented as a test to students learning algebra. Take a moment to read the following proof and do a self-test.

“Proof” that 1 = 2

Each of the steps seems to be correct, but again, the conclusion is wrong. Where is the mistake?



After solving that problem, tackle the next “proof” that demonstrates how 2 equals 3.

“Proof” that 2 = 3



Where is the mistake?

Why the proofs are wrong

In the first example, the proof breaks down at the step of dividing both sides by the term (a – b ). The proof begins with the assumption a = b , so the step of dividing by (a – b ) is the equivalent of dividing both sides of the equation by 0, which is a prohibited operation. The proof essentially has reasoning along the lines of “Since 2(0) = 1(0), which is true, dividing both sides by 0 leads to the conclusion that 2 = 1.” The paradox is resolved by removing the illegal operation.

The second example1 is wrong for a different reason. The mistake happens at the step of taking the square root of both sides. Before taking the square root, there is the valid expression that (2 – 5/2)2 = (3 – 5/2)2 , which simplifies to (-0.5)2 = (0.5)2 . After taking the square root of both sides, it is not proper to conclude the two quantities are equal. The correct conclusion is that either the two quantities are equal OR they are the opposites of the other (which is the case), and that resolves the apparent paradox.

While these “proofs” pose no problem to mathematics, they are interesting examples of why mathematical rules are important.

Notes

1. The example 2 = 3 is adapted from a paradox found on this website: http://www.clockstone.com/enigma/algebra_paradoxon_2.htm





7)The Least Interesting Number

What's the least interesting positive number in the world?

Well, it can't be 1 because 1 is a very important number: it is the smallest positive number and it is the multiplicative identity. The answer clearly can't be 2 since 2 is the smallest positive even number, plus it is the only even number that is a prime number. Then again, the answer also can't be 3 since 3 is the smallest odd prime number...

It would be easy to describe why the numbers from 1 to 100 are interesting. But there's really no point to continuing this process since there can't be a logical answer anyway!

Here is the reason why. One possible solution might be to conduct an election. One could nominate candidates for the least interesting number and then take a survey.

Suppose an online voting contest is conducted and that some number is called the least interesting number. Now that there is a winner, does that answer the question?

Not in the least. You see, any number that wins has the property that people considered the number to be the least interesting. Thus, this number won a contest, which is an accolade none of the other numbers have. That alone makes the winner at least a little bit interesting.

A possible solution is to disqualify the highest vote-getting number and crown the runner-up as the least interesting number.

But then again, if the runner-up gets the title, that again would make for a nice accolade, essentially negating the number's uninteresting nature. It must be the third place winner that should be considered...

You can see how this logic can cascade to eliminate the remaining candidates one-by-one.

Therefore, no number could possibly be the least interesting. And they say math is boring ?

A potential interview question?

Imagine a technical interviewer asks you the following question. What’s the least interesting thing about you?

How do you answer?

Many people might reply by explaining how “normal” they are in terms of physical appearance (“I’m average height”) or maybe in terms of a food preference (“I enjoy McDonald’s”).

In fact, the question is a sort of trick along the lines of the least interesting number paradox.

Imagine you could rank your attributes in terms of most to least interesting. Well, once you’ve come up with the least interesting aspect, one could argue that aspect itself must be interesting because it is the least interesting!

On a positive note, this logic suggests there is really nothing uninteresting about you at all. On the downside, such self-importance might explain the billions of boring updates on Facebook and Twitter.





8)Selected Self-Referential Paradoxes

Consider the rule: “All rules have exceptions.”

The rule seems fine. But the problem is if that rule is completely true, then that rule should also have an exception—that is, there should exist some rule with no exceptions. This conclusion contradicts the original rule.

The problem is the rule specifies a statement about “all rules.” In other words, the rule is referencing itself, or is self-referential .

A potential resolution is to modify the statement to be: “All rules have exceptions, except this one.” Now there is seemingly no trouble with the rule, unless you think very carefully about it.

Parents are well versed with the “rules have exceptions” paradox. A parent might tell a child never to lie, only to be caught lying the very same day. So to cover, the parent replies, “Do as I say, not as I do.” That sentence itself is not a paradox, but it is perhaps the more puzzling concept to understand.

There are many variations on the idea of a self-referential paradox; below is a sampling of them.

The Barber Paradox

This is one of the famous paradoxes by the logician Bertrand Russell.

Consider a barber who shaves everyone in the town except the people who shave themselves. The question is, who shaves the barber?

If the barber does in fact shave himself, that is not really allowed: he is only supposed to shave people who do not shave themselves.

If the barber does not shave himself, then he has classified himself as someone that he should be shaving.

Evidently the barber should and should not shave himself. What should he do ?

This seems to only be a logical problem and not a practical one, as no barber in the history of the world seems to have ever reported this dilemma.

The Liar Paradox

Try this one out for size: “This statement is false.” Is that a true or a false statement?

Note that if the statement is false, that means in fact it was a false statement, and hence the sentence was factually true.

On the other hand, if the statement is factually true, then that means the statement was wrong about being false. In other words, it was a false statement, which again is a contradiction to the premise.

In science fiction shows, the liar paradox always manages to foil evil logical villains or robotic computers who are uncomfortable with the contradictory logic. Humans apparently have no trouble with this sort of doublethink.

The Card Paradox

On a postcard there are written statements on each side.

Side A has the sentence: “The statement on the other side is true.” Side B has the sentence: “The statement on the other side is false.”

Which statement is true, if any?

One can work through the logic to see neither statement can be true without leading to a contradictory conclusion.

Pinocchio Paradox

This is a variation of the Liar Paradox that was created by an 11-year old who is the daughter of an Australian logician1 .

Pinocchio is a fabled wooden toy that becomes alive as a boy, with one special characteristic: his nose grows whenever he tells a lie. In the story, Pinocchio is caught telling a lie when he tries to conceal money, and eventually his nose keeps growing as he fibs .

Pinocchio's nose is a physical lie detector and that's good enough for story time. But what if Pinocchio were a logician or a computer programmer? Imagine the fun he could have if he tested his nose out.

Let's say he makes the innocuous statement along the lines of a Liar Paradox, “I am lying.” Now, if Pinocchio were truly lying, that would mean his statement is false, and he is actually telling the truth. But that would mean the statement was accurately described as a lie, so he was telling the truth. Would his nose grow or not?

There is an even cleverer thing he could say. Imagine that Pinocchio says, “My nose will grow.” If he is telling the truth, then his nose should not grow, which will on net make his statement false and make him a liar. If he is lying, then his nose will grow in response. But that would mean his statement was factual and so he was not in fact lying! It seems Pinocchio can trick his nose if he really wanted to.

“I Know Nothing” Paradox

This is a paradox attributed to Socrates.

If you state you know nothing, then you are claiming to be ignorant of any factual statements. If that were true, then your statement would be true and so you at least know that!

At best, you could state your ignorance as, “I know nothing, except this statement.”

Frightened Inmate Paradox

This is a scenario that takes place in one of my favorite TV shows, Arrested Development . In the episode “Marta Complex,” the character Tobias is auditioning for a role with the description “Frightened inmate #2.”

Tobias, a fledgling actor, is very worried about landing this minor role. In the plot, the narrator describes the paradox: “Tobias arrived at his audition for Frightened Inmate #2. The competition frightened Tobias, which he felt he could use in his performance. Unfortunately, this made him more confident, which frightened him again.” Poor Tobias .

This situation is closely related to the Paradox of Fiction . For fiction to be emotionally moving, the audience has to become attached to the characters and somehow think they are real. And yet, deep down, the story is known to be fictional. So how can an audience ever become attached to things that are known not to be real?

Clearly they can, as the tears after a heart-throbbing drama or the screams during a horror film attest. It might be because fictional stories lead to thoughts and reflection about things that that do exist and are genuinely sad or scary.

No Comparison

Have you ever heard someone say, “Oh, there is no comparison to this drink/cuisine/car/clothing brand?”

Now, if there were truly nothing to compare, then there would be no reason to make the statement “there is no comparison.” On the other hand, the fact there are things you can compare means the statement “there is no comparison” is false!

The expression of course means the item in question is of excellent quality and that one should not compare it to inferior products. But if that is the case, why not just say that?

Paradox of Tolerance

Tolerance is the idea that you can “tolerate” and respect people with other viewpoints. Tolerance can be a great virtue in that it means people of different races, religions, and opinions can peacefully get along.

But let's say that someone is bigoted and truly holds intolerant views. Will that person be tolerated by those who profess tolerance?

Absolutely not!

You see, tolerance is something that is intolerant of intolerance. You can't be tolerant towards everything, otherwise that means it is fine for people to be bigoted.

The paradox is somewhat captured in the quote, “I have seen great intolerance shown in support of tolerance,” by Samuel Taylor Coleridge .

This is a concept that was mocked in a South Park episode “The Death Camp of Tolerance” where the kids are sent to a camp to learn the importance of tolerance, only to learn that means they cannot tolerate intolerance.

Omnipotence Paradox

If a being is all-powerful, can it make a rock so big that it cannot lift it? If it can make such a rock, then it is clearly not all-powerful and hence not omnipotent. If it cannot make such a rock, then that again demonstrates a limitation, which again means the being is not omnipotent.

The paradox has been discussed for a long time because of its theological significance. Some people point out the flaw is in assuming such an omnipotent being exists, while others are content to clarify how an omnipotent being might be outside of humanity's logic system.

A similar paradox is the “irresistible force paradox” which wonders what would happen between an immovable object and an unstoppable force. This is often the philosophical dilemma faced between a villain and a super-hero, as neither can evidently achieve greatness without the other. This very paradox is stated by the Joker in the film The Dark Knight. Generally something gives, like the hero tricking the enemy, or maybe sacrificing himself to save the world. As for the resolution to the physical problem, perhaps the main issue is assuming that such impossible objects exist.

Notes

1. Peter Eldridge-Smith and Veronique Eldridge-Smith. “The Pinocchio Paradox.” Analysis (2010) 70(2): 212-215 first published online January 13, 2010 doi:10.1093/analys/anp173 http://analysis.oxfordjournals.org/content/70/2/212





9)The Debt that Must be Paid ?

The comedian Stephen Wright joked he once bought a humidifier and a de-humidifier for the purpose of setting them both in the same room to “fight it out.”

A similar conflict is present in the following paradox. Consider a student in the following situation: “My dad paid for my law school, offering that I could repay him after I won my first court case. After I took the loan and paid my tuition, he sued me for the entire amount.”

By a careful analysis of logic, the seemingly innocent loan sets up a no-win situation for the debtor, who faces two potential outcomes.

(A) Chances are the outlandish lawsuit would be thrown out, meaning the student would have won his first case. But in that case, he would be required to pay up per the original agreement.

(B) In the other case, if the dad somehow did win, that would mean the student has a legal obligation to pay the money back.

Then again, the student could equally argue the case means he does not have to pay. He could argue the following two circumstances.

(A) If he won the case, that would mean he has no legal obligation to pay.

(B) If he lost, then that would mean he still has not won a case, and hence, he should not have to pay!

Such a story is said to have happened in Greece between the philosopher Protagoras who offered a loan to his student Euathlus. (Protagoras was either greedy or Euathlus was a deadbeat, depending on which side you believe). Apparently the court was so confused that it adjourned for 100 years1 .

It might be a little bit arrogant, therefore, to speculate on what the correct answer should be. But let me do that anyway. In today's legal setting, the court ruling would probably take precedence. If the court decided the student did or did not have to pay, then the previous agreement could be annulled by the court's ruling. After all, the whole point of going to court is to try to settle the matter.

Notes

1. Suber, Peter. The Paradox of Self-Amendment. Available online. http://legacy.earlham.edu/~peters/writing/psa/sec20.htm





10)The Paradox of the Question

This is a delightful paradox written about by Ned Markosian in 19971 .

One day, an angel visits you and presents an offer of a lifetime. The angel offers to answer any question you wish to ask. You are allowed to ask about anything, but the angel emphasizes you can only ask a single question.

You realize the importance of getting the question right, and you ask around for advice. You post your situation on your blog, you tweet about it, you make a group on Facebook, and you email every single professor and smart person you know. Your post manages to blow up and suddenly everyone on Reddit is offering suggestions about the best possible question.

Some people ask practical questions (“What is the best diet to eat?”); some people make the question political (“How can we reduce gun deaths in America?”); and a few people ask questions for global good (“Can you tell me the cure for all diseases?”).

Some people think even grander, wondering to find out what the best possible question would be. But asking that question would be stupid, as it would waste the only question you are allowed to ask.

Ultimately, a mathematician proposes a neat idea that catches your attention. The person realized the logical trap of asking about the best question, and so the mathematician came up with a clever request to capture both the question and answer. The mathematician's question is: “Can you tell me the ordered pair (x , y ), where x = the best possible question to ask, and y = the answer to x .”

You think, this is brilliant! The question is just one question, and it sneakily asks for an ordered pair to learn about a question and answer.

You submit this idea on Reddit and it gets tremendously upvoted beyond every other option. Many professors, CEOs, and several Nobel Laureates agree this is the best possible question to ask.

Confident that you have done an exhaustive search, you finally get the nerve to ask the angel. You set up a live stream on your blog and billions of people tune in and anxiously await the answer.

After a moment, the angel offers the reply: “The answer to your question is that the best possible question x = the question that you asked me, and the answer to that question y = the response that I am giving to you right now.”

The angel disappears before your eyes, leaving you and the entire world stunned in silence. The question seemed to be perfect, and yet the angel's answer was so worthless that it provided no useful information to the world. You regret that you didn't just ask about the ending to The Sopranos . Even if you had asked a trivial question like, “What does 6 ÷ 3(1+2) equal?” you could have at least ended that Facebook debate of whether the answer is 9 or 1.

And yet, by asking the smartest question in the world—as the angel confirmed you did—you ended up learning nothing. The paradox is, how can asking the best question be wrong?

There are a few ways to resolve the paradox. One issue is presuming such an angel exists and would offer to answer a question. Remember this is a story so there is no real relevance. Another version of the story could have resolved with you learning the actual answer but then sleeping on it and forgetting it. The point is, it can be a mistake to over-analyze a hypothetical situation.

Another issue is the angel did not actually provide the best answer, since the best answer should benefit humanity in some way. Angels are supposed to do good things, not force people into logical traps.

A final issue is the question asked really was two questions. If you could ask for a pair (question, answer), then why not extend the logic to ask even more questions? Perhaps the question could be: “Can you provide me with the ordered n -tuple with the answers to how to end world hunger, cure all diseases, etc.”

Ultimately the problem is one of presupposing the scenario. It would be nice to have a supernatural angel answer an important question. For now, we have to settle with the solutions that people find .

Notes

1. Markosian, Ned. “Paradox of the Question.” Analysis 57 (1997), pp. 95-97. http://myweb.facstaff.wwu.edu/nmarkos/Papers/PQ.pdf

The format of the question (x , y ) is borrowed from Sider, Theodore. “On the Paradox of the Question.” Analysis 57 (1997): 97–101. http://tedsider.org/papers/paradox_of_question.pdf





11)Russell's Set of All Sets

This is a more technical paradox than the ones before, but it is an interesting one from set theory. A mathematical set is a collection of elements. For instance, the set S = {a , b , c } has three members, the letters a , b and c . The set S itself is an object too, and the set S is not a member of itself.

Consider another set T defined as “the set of everything not in S .” The set T will be very large, as it contains everything in the universe that is not the three letters a , b , and c . And one thing that T will contain is the set S . In other words, the set S is a member of T .

Going one step further, set T is not a member of the set S . Therefore, the set T is a member of itself. The whole point of this setup is to realize that some sets are members of themselves (like T ) while others are not (like S ).

Now comes the interesting part.

The set of all sets

The logician Bertrand Russell defined a set R as follows: “Let R be the set of all sets that are not members of themselves.”

The question is, is R a member of itself? Consider the logical possibilities:

(1) If R is not a member of itself, then R is a set that is not a member of itself. But by its definition, that necessarily means R must be included in R , a contradictory conclusion.

(2) If R is a member of itself, then R has the property that it is not a member of itself. This again contradicts the premise.

The logic breaks down because R is referring to itself. This paradox is a more general version of the Barber Paradox discussed previously.

One practical resolution—employed by standard set theory today—is that such sets R are not properly defined and so in effect they do not exist. If prohibiting certain sets sounds like a cop out, remember that other areas of math also preclude troublesome operations, such as the prohibition on dividing by zero. Developing the right rules is an important step to creating sensible mathematical conclusions.





12)A Paradox of Motion

This is one of the many delightful paradoxes of motion and time attributed to the ancient Greek Zeno.

Clearly motion exists as people experience the joy of movement. Yet Zeno made a clever argument of why motion should never happen.

Here is the argument. Imagine you want to run 1 meter. To reach the goal, you have to move half of that distance, which is ½ of a meter. Once at that point, you still have to move half of the remaining half-meter, which is ¼ of a meter. The logic can continue: at any point, you still have to move half of the remaining distance to reach the goal, so one then has to travel 1/8 of a meter, 1/16 of a meter, and so on. In other words, at any point, there is still space between you and the endpoint equal to half the remaining distance. Since there will always be some space, you will never reach the endpoint. Hence, motion is impossible.

Zeno's argument is baffling because it is logically reasoned and yet readily contradicted by physical experience. You can easily disprove the conclusion by standing up and walking one meter.

But the argument cannot be dismissed simply because its conclusion is wrong. One has to figure out the flaw in the logic.

The paradox can be resolved mathematically in an elegant proof using infinite series. Zeno's argument is that at each stage there is some halfway point that needs to be reached, and so there is always some “space” between the current point and the endpoint. Perhaps it is a mistake to focus on the “space” argument because it makes it seem there must always be a positive distance remaining. Mathematically, one could argue the spaces keep diminishing in size and get infinitesimally small, so small that eventually there is no space left.

But perhaps a clearer resolution is to focus on the distances traveled. Since you are traveling 1/2 of a meter, then 1/4 of a meter, and so on, we can equivalently add up the amount of distance traveled. If that distance adds up to 1 meter, then that means you can travel the distance without problem .

The infinite series of distances traveled is the following equation.

(I.) S = 1/2 + 1/4 + 1/8 + …

The hope is the series adds up to 1 meter. What is the value of S ?

The trick is to manipulate the sum algebraically by halving both sites. Doing this yields the following equation.

(II.) ½ S = 1/4 + 1/8 + 1/16 + …

Note that the infinite series in equation (II.) is precisely the same series in (I.) except that it does not include the 1/2 term. Therefore, if the equation in (II.) is subtracted from (I.), every term in the infinite series after 1/2 gets canceled out.

Thus, the subtraction of the two equations leads to the result ½ S = ½. The conclusion is that S = 1, which means the set of distances does add up to 1 meter.

Zeno's paradox is an example of how an infinite series can be used to satisfactorily resolve a logical argument with practical experience. This is not always the case as illustrated in the next two paradoxes.





13)How Can a Part be Equal to the Whole?

Galileo is attributed with this paradox about infinity. Now clearly the set of positive numbers 1, 2, 3, … and so on is infinite. For any number n , there is a larger number n + 1, meaning the sequence of numbers never runs out.

The question is, how big is the set of even numbers by comparison?

The intuitive answer is that the even numbers must be smaller. That is because they are a subset of all numbers, and a part should be smaller than the whole.

Galileo was someone who thought this was not correct. In fact, a careful mathematical argument can show the set of even numbers is equal in size to the set of all positive numbers.

One has to take a step back in order to make sense of this statement. What exactly does it mean to say that two sets have the same size?

In mathematics, the concept of size (or cardinality ) is a process of matching. Roughly speaking, the size of a set is determined by how it matches up with the set of positive numbers (or if it cannot match up, then it would be larger and uncountable).

A single person has a size of 1 because it can be matched to that number in a correspondence: single person ↔ 1

Similarly, a pair of people will have size 2 because the pair can be matched to the set of numbers {1, 2} as follows: person A ↔ 1, person B ↔ 2.

So when a set is said to be a particular size, it means the set can be matched in the process described.

Now, return to the question: how big is the set of even numbers? Using the matching principle, the even number 2 can be matched to 1, the even number 4 can be matched to 2, and in general, the even number 2n can be matched with the number n . This can be written out as: 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, …, n ↔ 2n, .. .

The list will continue forever, but the important detail is this: the two sets are matched up perfectly. Mathematically that means the two sets have the same size, or in technical terms, the same cardinality .

This example shows that infinity is a strange beast: a portion of the whole might still be of equal size to the whole.





14)Infinite Series Paradox

This problem1 is a bit more mathematical than the previous ones in this section. But it can be solved by with good reasoning skills and a little recollection of high school algebra.

See if you can figure it out.

Begin by defining the infinite series where the n th term is the fraction 1/n , as follows

(I.) S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

The next step is to take half of the series. That is, multiply both sides of the equation by one-half. The end result is the following series where every n th term is the fraction 1/(2n ).

(II.) ½ S = 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + ...

The next step is to subtract the series in (II.) from the series in (I.). This can be accomplished term by term. Since the second series has every even fraction 1/(2n ), and the first series has every fraction 1/n , subtracting (II.) from (I.) results in a series with only the odd fractions.

(III.) ½ S = 1 + 1/3 + 1/5 + 1/7 + 1/9 + …

Now for the final step. Both the series in (II.) and (III.) are different expressions for ½ S. Subtracting (II.) from (III.) yields the following nested series.

(IV.) ½ S - ½ S = 0 = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + …

Since we are subtracting ½ S from itself, the result should be zero. But each term in the nested series is clearly positive, and therefore the sum of the series is greater than zero.

How is this possible? In other words, what went wrong?

The harmonic serie s

The series S where the n th term is 1/n is also known as the harmonic series. What is its sum?

A good approach to problems like this is to work numerically. That is, add up some of the terms and see if there might be a pattern. The first ten terms (from 1 to 1/10) sum to about 2.92. The first 100 terms sum to about 5.2. The first 1,000 terms sum to about 7.5. Adding up a lot of terms, say the first 1043 terms, leads to a sum2 of just over 100. At this point, one might conclude the series sums to about 100. The reason is that the remaining terms are incredibly small fractions that would not seem to affect the sum.

Remarkably, that intuition turns out to be false. It can be proven the harmonic series does not converge. Even though the sum grows very, very slowly, like the Energizer Bunny, it keeps on going and going and going.

Here is a proof for why the harmonic series does not converge. Compare the harmonic series to a series where the first 4 terms are 1/4, the next 8 terms are 1/8, the next 16 terms are 1/16, and so on.

Harmonic Series : S = 1 + 1/2 + 1/3 + 1/4 + ….

Another Series : T = 1/4 + 1/4 + 1/4 + 1/4 + (1/8 + …) + …

The important thing to notice is the harmonic series has a sum at least as large as T . The reason is that the n th term in the harmonic series is at least as big as the corresponding n th term in the other series T (do a term by term comparison: 1 is bigger than 1/4, 1/2 is bigger than 1/4, and so on). So whatever the sum of T is, the harmonic series should at least be as large.

What is the sum of the series T ?

We know that there are 4 terms of 1/4, there are 8 terms of 1/8, there are 16 terms of 1/16, and so on. Therefore the sum of T is calculated as:

T = 4(1/4) + 8(1/8) + 16(1/16) + … = 1 + 1 + 1 + … = ∞

In other words, the series T has an infinite sum. And since the harmonic series has to sum to at least as large, that means the harmonic series also has an infinite sum. The sum grows very, very slowly, but remarkably the sum is unbounded.

Resolving the paradox

Here were the four steps that lead to the paradox.

(I.) S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + …

(II.) ½ S = 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + ...

(III.) = (I.) - (II.) ½ S = 1 + 1/3 + 1/5 + 1/7 + 1/9 + …

(IV.) = (III.) - (II.) ½ S - ½ S = 0 = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + …

Where is the mistake? Note that since the harmonic series has an infinite sum, that means half of the harmonic series also has an infinite sum. In other words, both the series in (I.) and (II.) have an infinite sum.

Therefore, one flaw in the proof is step (III.), which is the subtraction of two series that have an infinite sum. Since it is nonsensical to subtract one infinite quantify from another, the resulting steps are incorrect, and that's why step (IV.) is an absurd conclusion.

Notes

1. This paradox comes from the website http://www.clockstone.com/enigma/algebra_paradoxon_3.htm

2. Sloane's A082912 : Sum of a(n) terms of harmonic series is > 10^n, The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Available online. http://oeis.org/A082912 (Found via Wikipedia entry http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)





PROBABILITY PARADOXES

Think you understand randomness and chance? Take a stab at the probability problems in this section, which are famous for being both counter-intuitive and difficult to solve.

Some of the paradoxes demonstrate why it is a bad idea to rely on intuition (Birthday Paradox) .

In other times, the trouble is trying to understand why the correct answer makes sense, as seemingly immaterial details are relevant to find the odds (Bertrand's Box Paradox ).

A third set of problems is about the nature of expectations and random sampling (the St. Petersburg Paradox ).

Finally there are some problems that question the notion of probability and expectation (Random Chord Paradox) .





15)Birthday Paradox

Is it really surprising, or was it bound to happen? The birthday paradox illustrates why some surprising coincidences are not really all that surprising.

The setup is the following problem. How many people are needed so there is a 50-50 chance that two people will have the same birthday? (It is assumed that each person is equally likely to be born on the 365 days in a standard year).

When most people hear this problem, they conclude the group would have to be pretty large. The thought process might go as follows.

Linear thinking

On one extreme, the chance is very low in a small group. In a group of 2 people, for instance, there is just a 1 in 365, or less than a one percent chance, that the two people would share the same birthday.

On the other extreme, the odds become very favorable in a large group. In fact, at some point the group would be sufficiently large so that two people would necessarily share the same birthday. A little bit of logical thinking will demonstrate the crucial group size is 366 people. In any smaller group, it could be possible that each person had a distinct birthday. In a group of 366 people, at most 365 could have different birthdays. The extra person would necessarily have to share a birthday with someone.

Let's recap the range. If a group of 2 people corresponds to about 0 percent chance of a shared birthday, and a group of 366 people translates to a 100 percent chance, it stands to reason that a group size somewhere in the middle would correspond to a 50 percent chance. That is, one might guess a group of about 180 people would “cover” half of the days in the year, achieving the 50 percent chance of a shared birthday.

Now comes the surprise. The ballpark answer is not just wrong, but it is not even in the same ballpark. The birthday problem cannot be solved by considering the linear trend .

The answer

The solution is a mere 23 people are required to achieve a 50 percent chance of two people sharing a birthday. The apparent paradox is trying to reconcile why the linear intuition does not work, and why the actual answer is so low.

Why is the answer 23? We have to take a step back and think about what the question is asking. When considering birthdays, people naturally focus on their own birthday. Based on personal experience, you would not expect someone in a group of 30 or 100 people to share your birthday. That would be a surprise.

But that's not what the birthday problem is asking. The question is not asking whether you would share a birthday with someone. The question is whether any two people might share the same birthday.

The answer becomes intuitively easier to understand when considering the pairs of birthdays in a group. With two people, there is just one pair of birthdays to compare. But with 3 people, there are 3 pairs of birthdays: persons 1 and 2, 2 and 3, and 1 and 3. With 4 people, there are 6 pairs of birthdays that might coincide (as in (1, 2) (1, 3) (1, 4) (2,3) (2,4) (3,4)). You get the idea.

Each time the group size increases, that extra person could potentially match with each of the previous group members. By the time the group size is 23, there are a possible 253 pairs of people who could share a birthday. At this point, there is a 50 percent chance that two people would share the same birthday.

Here is the formal calculation. The first person's birthday can be any of the 365 days. So that the second person does not match, it would have to be one of the remaining 364 days. Therefore, the chance the two do not share a birthday is 364/365.

The third person's birthday cannot match either birthday of the previous two people. Hence, this person can only have a birthday on one of the remaining 363 days.

This logic can be extended for larger group sizes. More generally, the k th person in a group could have a birthday on one of the remaining (365 – k + 1) calendar days so as not to match with anyone else.

The joint probability that no one shares a birthday is the product of the individual odds. Hence, in a group of k people, the probability that no one in the group shares a birthday:

(364/365)(363/365)(362/365) … (365 – k + 1)/365

Since each of the terms has the same denominator of 365, and there are k – 1 terms, this expression can be rewritten as:

(364)(363)(362) … (365 – k + 1)/365k -1

The chance that someone has the same birthday is the complement event, which is found by taking 1 minus the previous equation:

1 - (364)(363)(362) … (365 – k + 1)/365k -1

The chance that two people share a birthday is still small for small groups (low values of k ). In a group of 5 people, there is a 2.7 percent chance that two people will share a birthday. In a group of 10, this rises to a noticeable level of 11.7 percent. For each of the next 10 group sizes, each person added to the group increases the probability by about 3 percentage points. As the group gets larger, it becomes harder to avoid sharing a birthday; or in other words, it becomes more likely that two people will share a birthday.

In a group of 23, there is just over a 50 percent chance that two people will share a birthday.

Now, part of the paradox is that, in a group of 23, you in particular will not have a 50 percent chance of someone sharing your birthday. It is just some two people in the group will share a birthday.

The late-night TV host Johnny Carson famously misinterpreted the birthday problem, as detailed in John Allen Paulos' book Innumeracy . Carson noted there were over 100 people in the audience—much more than 23—so he concluded someone in the audience must have his birthday. Evidently no one did, which made the viewer question the birthday problem answer .

But Carson was wrong in what he asked. If he had surveyed the audience to find if any two people shared a birthday, the result would have been almost surely yes. Instead, he was focusing on his particular birthday, which is an entirely different question.

How big a group is required so there is a 50-50 chance of sharing a specific birthday? This problem is a bit easier to solve and matches the intuition that a large group is needed.

The chance someone has your birthday

There are 364 birthdays that are not yours, so in a group of k people, the probability that none of the people shares your birthday is (364/365)k . In other words, the chance that someone shares your birthday is 1 – (364/365)k . This equation can be set equal to 50 percent and solved to find that k should equal 253. In other words, you need at least 253 people so that someone matches your specific birthday with a 50 percent chance.

So while it might be relatively unusual that someone has your exact birthday, it is relatively unsurprising in a large group that two people would share a birthday.





16)False Positive Paradox

Suppose a rare disease afflicts 1 percent of the population, and the test for it is 99 percent accurate.

Fearing an outbreak, public health officials mandate tests for everyone in a town of 100,000.

The question is: if a person tests positive, what are the chances the person actually has the disease?

Because the test is 99 percent accurate, it might seem the results should be accurate too. In fact, the results are expected to be quite inaccurate: someone who tests positive will in fact have a 50 percent chance of not having the disease.

The question is, why does a very accurate test produce such inaccurate results?

Resolving the apparent paradox

The issue is the disease is rare. So even though the test will accurately identify people who have the disease, it will inadvertently misdiagnose healthy people as false positives. Here is the supporting calculation.

Of the 100,000 people in the town, the disease affects 1 percent, or 1,000 people. The test, which is 99 percent accurate, will identify 990 of the group as positive.

What about the healthy people? Of the 99,000 healthy people, the test will be 99 percent accurate. That means the test will identify 98,010 of the group as healthy. That leaves 1 percent, or 990 cases, that will be wrongly identified as false positives.

To summarize, the test will yield 1,980 positives: of which, 990 are correctly identified, and 990 are false positives.

So what does that mean? A large portion of people who tested positive—50 percent (990/1,980)—were false positives.



While the test is 99 percent accurate, someone who tests positive only has a 50/50 chance of having the disease. The result is a consequence of the disease being rare. While the test can correctly identify people who have the disease, it will also misidentify many healthy people as false positives.

What the paradox teaches us

The problem illustrates why it might be necessary to run a test, even an accurate one, a second time. Using the same numbers above, if the test were run a second time, the 50 percent false positive rate would drop to just 1 percent. (Of the 990 false positives, only 10 would test as false positives a second time. In contrast, 980 of the 990 correct positives would test as positive a second time. So only 10 of the 990 people who tested positive two times would be false positives. This is a 1 percent rate of false positives. This level of accuracy is more in line with the stated 99 percent accuracy rate.)

Second, the problem suggests why mandatory screening could be a bad idea. For instance, the federal government could require that every couple get tested for various diseases before marriage. But given the millions of marriage every year, this blanket policy would probably result in an overwhelming amount of false positives, scaring many people for no reason.

What's the alternative to testing everyone? Doctors are cautious not to run a barrage of tests on everyone. They take medical histories and look for symptoms. They often administer tests to confirm their suspicion, not to screen for a problem. This is one reason why medical tests are often useful: it's the doctor who does some initial screening to decrease the rate of false positives.





17)Simpson's Paradox

This paradox is often explained with hospital survival rates. I discovered another example in America's college football rankings called the BCS ranking system.

Without getting into too many details, the BCS ranking broadly works like this. Teams are ranked independently by 3 different voting systems: the Harris Poll, the USA Today Poll, and a computer algorithm.

The 3 polls are then averaged, with each poll given equal weight. The final number, known as the BCS average, determines the overall ranking of teams.

The average is supposed to even out the biases of the various rankings. But several years ago I noticed something rather peculiar in the rankings.

Namely, I found this: a team that was ranked 6th or worse in all three polls ended up being ranked as the 5th overall team after the polls were averaged.

The weird thing I noticed occurred for the BCS rankings from October 26, 2008.

At the time, USC was ranked 7th according to the Harris Poll, 6th according to the USA Today Poll, and 6th according to the Computer Rankings.

And yet when all three rankings were averaged, USC ended up being ranked 5th overall. Here is the detailed table with the rankings1 .



Here is a brief explanation of some of the numbers in the table:

–USC was ranked as 7th in the Harris Poll. Its Harris Poll average was 0.7912, behind Florida 6th (0.7923) and Texas Tech 5th (0.8021)

–USC was ranked as 6th in the USA Today Poll with an average of 0.8052.

–USC was ranked as 6th in the Computer Rankings with an average of 0.750.

–The three polls were averaged together to determine a BCS average. USC was the 5th highest number at 0.7822. The BCS average was calculated as the average of the three rankings, that is, 0.7822 = (0.7912 + 0.8052 + 0.750) / 3

What is going on?

If USC was ranked no better than 6th in any individual poll, how could it have jumped up to 5th overall when the three polls were averaged?

The resolution

Basically USC was behind Texas Tech (and close to Florida) in the Harris and USA Today polls .

USC got a boost because its computer ranking–though it was 6th–was a large value. This basically let it leapfrog Texas Tech and Florida who got punished by the computers.

Here's an illustration of how USC had virtually the same ranking in other polls, but a large boost from the computer rankings.



The paradox is in thinking that each poll actually affects the overall ranking equally. This is not true: it is the total points that affect the overall average. Because the computer rankings punished Texas Tech, this skewed their overall average and dropped them behind USC.

Other examples

Simpson's paradox happens when combining averages for groups of different sizes. A canonical example is that one hospital can have better survival rates for cancer and heart patients, but it might be worse when looking at overall survival rates for both cancer and heart patients. The reason is the same as the football example: combining groups of different sizes can alter the overall average.

Similarly, it is theoretically possible for a baseball batter to have the highest average in both first and second halves of a season, but lose out to someone else in the overall batting average for a season.

Notes

1. The BCS rankings data were originally from the website ESPN.com. A cached copy of that data, with USC's data highlighted, is available here: http://mindyourdecisions.com/blog/wp-content/uploads/2012/01/bcs-statistical-paradox.png





18)Bertrand's Box Paradox

Imagine I have three drawers. One drawer is filled with 2 gold coins, another is filled with 2 silver coins, and a third is filled with one of each coin.

Now I randomly pick a drawer from which I randomly select a coin. It turns out to be a silver coin. The question is, what's the chance the other coin is also a silver coin?

The paradox: it's not 50 percent!

Common sense is that the other coin could either be a silver coin or a gold coin, hence the answer should be 50 percent. But if that were the case, this problem really wouldn't be very interesting, and this wouldn't be a paradox.

The answer is the other coin has an overwhelming 2/3 chance of being a silver coin. Huh? How come looking at a coin makes the other coin more likely to be the same type? We have to carefully consider the information we are given to see why that is the case.

Resolving the paradox

The trick is to carefully account for the possibilities. Let's label the coins as G for gold, g for a second gold coin, S for silver, and s for a second silver coin.

The logic is as follows: I begin by picking a drawer at random from the possibilities Gg , GS , and Ss . When I select a coin at random and it's silver, you can eliminate the possibility that I drew from the Gg drawer.

There are three equally likely events to consider.

(1) I am showing S from the Ss drawer, and the other coin is s .



(2) I am showing s from the Ss drawer, and the other coin is S .



(3) I am showing S from the SG drawer, and the other coin is G .

In two of the three cases, the other coin is silver. Hence, the probability the other coin is silver is 2/3 .

The trick is that when you see a silver coin, it could really be one of two of the indistinguishable silver coins in the Ss drawer.

If you understand this problem, then the next paradox should be a cinch.





19)Boy or Girl Paradox

You're doing a statistical analysis for the government. One of your co-workers is looking at a set of 10,000 parents who happen to have two children, with at least one boy.

Your co-worker poses a problem to you, “I picked a random family from the data set. What is the probability that both children are boys?”

(Another way this story is told. A stranger tells you she is the mother of 2 children. At least one of the children is a boy. What's the chance that both are boys?)

The paradox: it's not 50 percent!

This is one of the stranger paradoxes. We know the genetic odds of having a boy or girl are roughly 50/50. If someone tells us that one of their children is a boy, why should that affect our belief about the gender of the other child?

The answer is stunning because the information revealed changes the odds. It turns out there is just a 1/3 chance that both are boys.

Resolving the paradox

A parent of 2 children could either have four possible scenarios of younger and older siblings, with B for boy and G for girl: BB, BG, GB, GG.

When you learn that at least one of the children is a boy, you can remove the GG possibility. Hence, there are three equally likely events to consider:

(1) the parent has two boys



(2) the parent has a boy with an older sister



(3) the parent has a boy with a younger sister

In two of the three cases, the other sibling is a girl. Hence, the probability the child is a boy is 1/3 .

I admit I found this solution hard to believe at first. So I ran a test in my spreadsheet using a random number generator to check. I created a fictional data set of 10,000 families who had two children and at least one boy (I did this by creating about 14,000 families with two children and removing the ones with two girls.). I then counted how many of the families had two boys. The answer was 3,346, which means the chance that both children are boys is 33.46%, which is very close to the theoretical answer of 1/3.

What this paradox teaches us

This is a somewhat controversial paradox as it depends on the exact phrasing. If I wrote, “the older child is a boy,” then we'd consider two possibilities:

(1) the parent has a boy with a younger sister



(2) the parent has a boy with a younger brother

These two events are equally likely, and so the answer would be 1/2. In this paradox, it's the phrasing that is meant to be tricky, not the math.





20)The Monty Hall Problem

This is perhaps one of the most famous probability puzzles. The problem gained notoriety when a reader Craig Whitaker posed it as a question to the genius Marilyn vos Savant in Parade magazine1 .

The question is the following. Suppose you're on a game show called Let's Make a Deal! There are three doors; behind one is a car, and behind the other two are goats. You pick door 1, and the host opens door 3 to reveal a goat. The host asks if you would like to switch to door 2. Is it better to switch or stick with your initial choice?

Incidentally, the problem is named after the host of the show, Monty Hall.

The paradox: the odds are not even

We need to specify a few circumstances before we can solve this problem. We will assume that the prizes are placed behind each door at random, so that there is a 1/3 chance that the car is behind each of the doors. We also need to make an assumption about the host's behavior. The host is not going to open a door and show you there is a car behind it. That would kill the suspense of the moment since it would mean you are either staying on a door that has a goat or switching to a door that has a goat. That's not very good TV, is it?

Even with this setup, it seems like it shouldn't matter if you switch or stay. After the host opens one of the doors and shows a goat, that means one of the doors has a car and the other has a goat. It would appear that there is a 50/50 chance that either door could have the car, and hence it doesn't matter if you switch or stay.

This reasoning sounds right but it is wrong. Remarkably you have a 2/3 chance of winning if you switch. The reasoning is along the lines of Bertrand's Box Paradox . The idea is that the host opening the door does reveal some information.

One line of logic can demonstrate the soundness of switching. Note that if you stay, you only win if the initial door you picked had the grand prize. This is a 1/3 chance. On the other hand, if you switch, you would win if the grand prize were in either of the doors you did not pick . That's a 2/3 chance.

Another way of seeing this is to write out the three possible scenarios and seeing which strategy wins. If:

–Grand prize is in door 1 (1/3 of time): you win if you stay on door 1.

–Grand prize is in door 2 (1/3 of time): the host will open door 3. You win if you switch but you lose if you stay on door 1.

–Grand prize is in door 3 (1/3 of time): the host will open door 2. You win if you switch but you lose if you stay on door 1.

We can add up these contingencies to conclude that you will win 2/3 of the time if you switch, but you only win 1/3 of the time if you stay.

Consider another thought experiment. You pick door 1 from doors 1 to 100. The host is ready to open 98 of the doors that you did not pick to reveal a goat behind each door. If you stay on door 1, you only win in the 1 in 100 chance that door 1 had a goat. If you switch, then you would win if the goat were on any of the 99 doors you did not pick.

The situation is paradoxical because it initially seems that the host opening the door doesn't change anything. The reality is that the host never opens the door with the grand prize. Hence you learn information when he opens the door, and you gain an edge by switching.

The problem is also a bit of a stain on the mathematical community. After Marilyn vos Savant wrote up the solution, many readers did not believe it was a 2/3 chance. People wrote thousands of letters, including some people who had a Ph.D. in mathematics, that said she was wrong and the answer was 1/2. And partly for this reason, the Monty Hall problem joined the ranks of famous math problems.

Notes

1. The problem appeared in the magazine Parade in 1990. Marilyn vos Savant has the problem and several critical letters listed her webpage. http://marilynvossavant.com/game-show-problem/





21)The Necktie Paradox

Two men are given neckties by their wives as Christmas presents1 . Over drinks they start arguing about who has the cheaper necktie.

They decide to make a playful wager to settle the matter. They will talk to their wives to find out which necktie is more expensive. The man with the more expensive necktie will be the loser, and additionally, he has to hand over his necktie as a prize.

How fair is this wager?

Is the other necktie always better?

The first man is thinking about the bet, and he suddenly feels that he has made a good wager.

He reasons as follows: “the probability of me winning or losing is an even 50/50 chance. If I lose, then I lose the value of my necktie. If I win, however, then I win more than the value of my necktie. That is, I can bet something worth x dollars and have a 50 percent chance of winning something valued more than x dollars. It's a winning proposition, and I should definitely make this wager!”

The second man, also thinking about the bet, uses the similar reasoning to arrive at the same conclusion. He feels the bet is a winning proposition, and he is happy to take the wager.

It is obviously not possible that both men could be advantaged in a zero-sum game. What is the reason for the necktie paradox?

Resolution 1: correct the logic

One of the problems is the men are reasoning incorrectly, a common downfall when trying to do math over drinks.

The issue is that each is considering the wager in terms of his own tie being the more expensive and less expensive tie at the same time . Clearly, one's own tie has to be one or the other, either the more expensive or the less expensive .

Therefore, the reasoning should be:

–If I have the more expensive tie, and I make the bet, I will lose my more expensive tie, or

–If I have the less expensive tie, and I make the bet, I will win the more expensive tie

There is a 50/50 chance of being in either situation. Hence, there is no advantage to either player, and there is no mystery to the bet.

The game will end with each man winning half of the time, and hence the bet is fair.

Resolution 2: correct the expectation calculation

There is another way to view the problem. Suppose the more expensive necktie is worth x dollars. What is the expectation of the bet?

To calculate the expectation, you need to know two things.

First, what is the chance you win? This is the chance that your tie is more expensive. It is assumed that neither you nor the other man knows the value. Under such ignorance, it is reasonable to say there is a 50/50 chance that either tie is more expensive. Hence you win with a 50 percent chance and lose with a 50 percent chance.

Second, what is the payoff to the bet? In the case you win, you win the more expensive tie worth x dollars. In the case you lose, you have to give up your more expensive tie worth x dollars.

Putting these two facts together, we get:

E(bet) = Pr(win)gain + Pr(lose)loss = 0.5(x ) + 0.5(-x ) = 0

The math demonstrates the wager has an expected value of zero, that is, the bet is exactly fair to both players.

Resolution 3: never bet a gift from your wife

Of course, it should be clear the game is not really a zero-sum game but a negative sum game .

Upon learning the bet, and that their husbands would wager their thoughtful gifts, both wives will be angry. Clearly, there will be no winners, and the only safe bet is to avoid this game entirely. Consider yourself forewarned!

Notes

1. This description of the problem comes from the Wikipedia entry. http://en.wikipedia.org/wiki/Necktie_paradox





22)The Friendship Paradox

It’s a mathematical fact. Your friends are probably more popular than you are.

Don’t believe it? Consider this: the average Facebook user has 245 friends, but the average friend on Facebook has 359 friends, according to a 2011 Pew survey1 .

That’s right. The average person on Facebook has fewer friends than their friends do.

But how can that be? It definitely seems weird, and that’s why this phenomenon is known as “friendship paradox,” described in a 1991 paper by Scott L. Feld amusingly titled “Why Your Friends Have More Friends Than You Do.”2

The paradox is not so mysterious, however, after examining its emergence from mathematical facts.

The intuitive explanation

One way to think about this is to remember when you first joined Facebook. When you first created a profile, you started out with 0 friends. The quickest way to get friends was to add people who were already on Facebook. And since these people were already using Facebook, they already had a huge lead in friend count. And each time you added a friend, those people got to increase their friend count too! So clearly, it’s not that hard to see why your friends would have more friends than you when you first joined.

Of course, as you invite more friends and make contacts you can certainly overtake people in the friend count. The friendship paradox is not about the time you joined.

It’s about this: on average, you will tend to add friends of people who are popular because it’s a social game.

And in the end, the average Facebook user ends up having fewer friends than their friends do .

A concrete example

Let’s consider a group of 4 people A, B, C, and D.

Imagine that A and B are friends, B is friends with everyone, C is friends with B and D, and D is friends with B and C.

We can draw a graph that illustrates the various friendships, where a connecting line means two people are friends:



Let’s go through the calculation of counting friends, and counting friends of friends, to see if the friendship paradox holds.

A has just 1 friend, B has 3 friends, C and D each has 2 friends.

Now we will count the friends of friends. A is friends with B who has 3 friends. B is friends with everyone, which makes for 5 friends of friends. Similarly we will find C and D each has 5 friends of friends.

Here’s a table that summarizes these statistics:



You can see the average friends of friends column is higher for everyone but person B. So B is the lone exception: a very popular person will have more friends than their friends do (not a shock at all).

But now comes the interesting part when we look at the average. When we add up the total number of friends, we find there are 8 total friends amongst the 4 people. This means the average person has 2 friends.

On the other hand, we can find the total in the friends of friends column. Here, we will find there are 18 total “friends of friends” to be divided by the total of 8 “friends” relationships. This means the average friend has 2.25 friends.

In other words, the average person has 2 friends but the average friend has 2.25 friends. This is the friendship paradox.

Now let's prove why this happens.

The mathematical proof

We need to calculate two quantities: the mean number of friends for the entire graph, and the mean number of friends for a random person/node on the graph.

A little bit of notation can help. Let us say that person i has a number of friends x i , and suppose there are n people.

What is the average number of friends in the entire graph? This is easy to calculate. We add up the total number of friends and divide by the total number of people. The formula will be:



We don’t need to simplify that formula any more for our purposes. Now, we need to calculate the average friends of friends. This will be a bit harder, but not too bad.

The way we will do this is as follows. For each person, we will look at their friends and then add up the number of friends they have. So if B is friends with A, C, and D, we will need to add the numbers x A , x C and x D .

The formula is a bit messy if you work person by person. Luckily, there is a trick.

We really only care about the total sum since this is an average. So we can ask the following question. For a given person i , how many times will the term x i appear in our final summation?

The answer is easy. The only time we need to count friends of person i is for the friends of person i (I know that sounds confusing, read it again and it will make sense).

That means each of the friends of person i (there are x i such friends) will contribute the term x i to the final sum.

Hence, the final summation has the term (x i )(x i ) = x i 2 .

The numerator will therefore be the sum of the squares of each person’s friends. We need to divide this by the total number of friends which is the sum of x i .

Therefore, the mean number of friends OF friends is:



Again, we do not need to simplify this term for our purposes.

What we will do is tinker with this formula so that we can compare this average to our previous one. The motivation is to rewrite the numerator with the squared sum in terms of other variables.

Once more, a statistical trick is involved. One has to remember the formula for variance, denoted σ2 , of a discrete random variable is the following, where μ is the population mean:



We can manipulate this equation to solve for the sum of the squared terms:



And now, we will substitute for the sum of the squared terms in the expectation we wish to compute:



We have found the average friends of friends is equal to the average friends PLUS another term. The other term is the variance of the number of friends divided by the average number of friends–both of these are non-negative in a typical social graph.



So in the end, we have proven the average number of friends of friends will be equal or exceed the average number of friends.

An application: predicting who gets the flu

Researchers have found a way to apply the friendship paradox to the flu .

From a public health standpoint, one wants to vaccinate everyone to lessen the spread of the flu. But we know that is typically not possible. So another approach is to target people who are most likely to spread it. This will be someone who is social and has lots of friends.

Ideally, we could draw a graph of all the friendships and see who is a prime candidate. But in practice, that will be expensive and take a lot of time and money.

The researchers came up with this idea3 . First, take a random sample of people (like taking random nodes in a graph). Then, ask those people to name a friend.

Chances are, those people will not name an outcast. They will name someone that is popular. And by the above analysis, that person will likely be more connected and more likely to get the flu.

The study found that friends of the random sample did in fact get the flu earlier. This indicates friends of friends could be prime targets for getting vaccinated.

Notes

1. Number of Facebook friends: Pew Survey via The Washington Post

2. Why Your Friends Have More Friends Than You Do. Scott L. Feld. American Journal of Sociology. Vol. 96, No. 6 (May, 1991), pp. 1464-1477. Published by: The University of Chicago Press. Article Stable URL: http://www.jstor.org/stable/2781907

3. Infectious personalities , The Economist.





23)St Petersburg Paradox

This chapter also appears in Math Puzzles Volume 1 .

You are offered an unusual gamble.

A fair coin is tossed until the first heads appears, which ends the game. The payoff to you depends on the number of tosses. The payoff starts at $2 and doubles on each successive toss.

That means you get $2 if the first toss is a head, $4 if the first toss is a tails and the second is a heads, $8 if the first two tosses are tails and the third is a head, and so on. In other words, you get paid $2k , where k is the number of tosses until the first heads.



The question is how much should you be willing to pay to play this game? In other words, what is a fair price for this game?

The paradoxical part

The typical question is to compute the expectation. This is done by multiplying the various payouts by their probability of occurrence and adding it all together. To say it another way, the expectation is the weighted sum of the payouts given their likelihood.

The respective probabilities are easy to compute. The chance the first toss is a heads is 1/2, the chance the first toss is a tails and the second is a heads is (1/2)(1/2), and the third toss being the first heads is (1/2)(1/2)(1/2), so the pattern is clear that the game ending on the k toss is (1/2)k

Now, with probability 1/2 you win $2, with probability 1/4 you win $4, with probability 1/8 you win $8 and thus the expectation is:

E = (1/2)($2) + (1/4)($4) + (1/8)($8) + … = $1 + $1 + $1 + … = $ ∞

The surprising result is the expectation is infinite. This means this game–if played exactly as described –offers a potentially unlimited payout. In theory, a rational player should logically be willing to pay an astronomical amount to play this game, like paying a million dollars, a trillion dollars, and so on.

The fair price of infinity is paradoxical because the game at first does not seem very favorable. Few people would be willing to pay more than a couple of dollars to play this game, let alone $100 or $1,000.

But expectation theory seemingly says that any amount of money is justifiable. Banks should be willing to offer loans so people could play this game; venture capital firms should offer more money than they do to start-ups; individuals should be willing to mortgage their house, take a cash advance on their credit card, and take a payday loan.

What’s going on here? Why is the expectation theory fair price so different from common sense? It turns out there are a variety of explanations.

Resolution 1: Payouts should be realistic

Imagine you are playing this game with a friend. You hit a lucky streak. The first nine tosses have been tails and you’re still going. If the tenth toss is a heads, then you get $1,204 as a payout. If it’s a tails, you have a chance to win $2,408, and even more.

At this point your friend realizes he’s made a mistake. He thought he’d cash out with your $10 entry fee, but he now is in trouble and he cannot afford to pay any more.

He pleads with you to stop. He’ll gladly pay you the $512 you’ve earned–so long as you keep this whole bet a secret from his wife. What would you do in this situation?

Most of us would take the cash and show some mercy. There is no joy in winning if it means crippling a friend financially. And this concocted scenario leads to one of the unrealistic assumptions of the St. Petersburg Paradox.

In the hypothetical coin game, you’re supposed to believe the other side can pay out infinitely large sums of money. It doesn’t happen often, but if you get to 20 coin tosses, you fully expect to be paid $1,048,576.

This is unrealistic if you’re playing with a friend or even a really, really rich friend. It might be possible with a casino, but even a casino may have a limited bankroll.

The truth is that payouts cannot be infinite. If such a game were to exist, there must be a maximum, finite payout.

This means the expectation is not an infinite sum but rather a sum of a finite number of terms. Depending on the size of the bankroll, the gamble has a finite payout.

Here are a few calculated expected values when considering a bankroll with a maximum payout1 :



Even in the unlikely event you are playing against Bill Gates, who could pay out an amazingly high amount, the gamble could not possibly be worth more than $37.

As you can see, with a more realistic model of the game, the gut feeling that the game is worth a few dollars matches up with the math of the gamble.

Resolution 2: diminishing marginal utility

A quantity like $1,000 has meaning to most people. If you were to ask a friend for such a loan, they would ask how you can pay it back, what you would use it for, and so on.

But there are times when $1,000 seems to lose its value. I like to think about the show Deal or No Deal where contestants play a multi-stage lottery to win $1,000,000. At various points in the game the contestants can either keep pursuing the big prizes or they can accept smaller consolation prizes. As the prospect of a big prize increases, the contestants start to care less and less about smaller prizes like $1,000.

This is an example of the famous concept of diminishing marginal utility–the idea that at larger levels of consumption, incremental units are worth less. The concept is applicable for wealth decisions because at some point incremental earnings mean less to a person.

What this means for the gamble is that the payouts should be altered. The payouts should not be measured in dollars but rather by the utility the money provides.

One way to model utility is to use a logarithm function. Instead of saying the payout for the first toss is $2, we will say it is log(2) units of utility, and accordingly adjust a payout of $x to be a utility of log(x ).

Using a log utility function, the St Petersburg game now has a finite payout. Here is a derivation:

E = (1/2) log 2 + (1/4) log 4 + (1/8) log 8 + …

E = (1/2) log 2 + (2/4) log 2 + (3/8) log 2 + …

Let u = log 2. Then ,

E = (1/2) u + (2/4) u + (3/8) u + …

- (½) E = - (1/4) u - (2/8) u + …

Therefore,

(½) E = (1/2) u + (1/4) u + ….

½ E = u

E = 2u = 2 log 2

This is a small payout but the actual quantity does not matter. The point is the payout is finite, which again resolves the original problem.

The St. Petersburg Paradox is interesting in that a simple game could yield a mathematical infinite payout. But in reality, the game would have a finite expected value. Thus, this matches the intuition that one should only pay about a fistful of dollars to play this game.

Notes

1. The table with maximum payouts for a given bankroll was adapted from a table on Wikipedia http://en.wikipedia.org/wiki/St._Petersburg_paradox





24)Wild Card Poker Paradox

Wild card poker is a variation of poker that implements the use of at least one "wild card." A wild card is a designated card that a player can assign any value and suit. A wild card livens the action by making stronger hands easier to complete.

This section is about the following question. In wild card poker, which hand is more valuable: three of a kind or two pair?

It turns out the question is something of a paradox. Before I get into this question, it is useful to understand the logic of poker hand rankings.

What makes a poker hand valuable?

Poker hand rankings could be arbitrarily chosen and simply based on tradition.

But, as everyone knows, there is a clever logic to poker rankings. The value of a hand is determined by its mathematical probability of occurrence. A hand is ranked more valuable if it is less likely or harder to happen by chance.

This is why a flush has a higher ranking than a straight: it is harder and less probable to complete five unsequenced cards of a suit (flush) than to get five sequenced but unsuited cards (straight).

It is a standard probability exercise to verify that the poker hand rankings are based on their frequency of occurrence. Here is a table that summarizes the rankings and probabilities1 :



In a way, the frequency ranking makes the game of poker fair and logical. A three of a kind is more valuable than a two pair exactly because it is harder to achieve by about two percentage points.

This ranking is fundamental to standard games of poker. But an extension is to ask: what happens when a wild card is introduced?

Wild card poker effects

Poker can also be played with wild cards to liven up the action. One common variation is to add a joker card to the deck and allow it to be wild and represent any card.

The addition of a wild card does three things.

1. It creates a new hand "five of a kind" where all five cards are the same value, like five aces or five kings.

2. It affects the probability of getting a hand. It is now easier to make stronger hands because the wild card can be designated to be any card. A player with a wild card can always, for instance, make a pair .

3. It forces a player to make a choice about the wild card. Which card should the wild card represent? The answer is obviously to make the choice that gives the best possible hand with the other cards.

This third effect creates a new choice for the player in declaring a hand. If a player holds a pair and also a wild card, then there are two possible ways to declare the hand. The hand can be called a three of a kind if the player declares the wild card to match the pair. Alternately the hand can be called a two pair if the player declares the wild card to match an unpaired card to go along with the pair.

(Say a player has 4,4,2,6, and a joker. The joker can either be a 4 to complete a three of a kind 444, or it can be a 6 to complete two pairs of 44 and 66)

Which hand should be chosen? This is the source of the paradox.

The wild card poker paradox

For the moment we will assume the standard ranking that three of a kind is more valuable than a two pair.

We can then calculate the probability of making each hand in wild card poker. We will make the assumption that a player holding a pair and a wild card will choose to make this hand a three of a kind rather than a two pair.

The resulting probabilities are2 :



Most of the frequencies are in line with the standard ranking of poker hands, but there is one glaring exception. The three of a kind occurs with a slightly