Paradoxes in probability

Many of the following questions, which are simple, have several different "reasonable" answers.  Depending on what assumptions you make, most of the questions have several distinct correct answers, too.  Your task is to find answers you can defend and to identify the assumptions needed to justify your answers.  Please be prepared to present and justify your answers in class next week.

The Monty Hall problem

Game show host Monty Hall ("Let's Make a Deal") asked winning contestants to select one of three closed doors.  Behind exactly one of the doors was a valuable prize.  Occasionally, after the contestant had selected a door but before it had been opened, Hall would open a different door to show it did not contain the prize.  He then allowed the contestant to change her choice to the remaining closed door.  Assuming Hall did not know which door contained the prize (and therefore needed an assistant to identify the door to open, to avoid inadvertently revealing the prize), what is the best strategy: switching or not switching (or does it not matter)?

The envelope problem

You are given two sealed envelopes.  In each is a sum of money.  You are told that one envelope has exactly twice as much money in it as the other.  You are allowed to open one and count the money.  Then, before the other envelope is opened, you make a decision: you may keep the money or you may take the money in the other envelope.  What is the best strategy: keep the money or switch envelopes?

The airplane puzzle

The famous statistician Abraham Wald studied hits on aircraft returning from combat during the second World War.  A pattern was clear: the rate of hits near the engines was notably lower than elsewhere, such as in the fuel system.  The military, which was interested in protecting the most vulnerable parts, therefore decided that shoring up the engine armor should have lower priority than protecting the rest of the plane.  What would you recommend and why?

The three coins problem

A hat contains three quarters.  One is normal, one has two heads, the other has two tails.  One quarter falls out of the hat onto a table, heads up.  What is the probability the other side is also a head?

The sibling mystery

A boy you meet on the street tells you he comes from a family of two children.  What is the probability he has a sister?  (What assumptions are needed for this question even to make sense?)

Lewis Carroll's problem

A bag contains a bean that is known either to be white or black.  A white bean is added to the bag, the bag is shaken, and one bean is taken out at random.  It is white.  What, now, is the chance that a bean taken out of the bag will be white?

Bertrand's paradox

What is the probability that a "random" chord in a circle is longer than the circle's radius? 

The bear cubs problem

The zoo has two bears: a white one and a black one.

What is the probability that both bears are male?
The zookeeper comes out and tells you that Bert is feeling better today.  Evidently, one of the bears is male.  What is the probability that both are male?
You ask the zookeeper which one is Bert and the keeper points to the light bear.  What now is the probability that both of the bears are male?

The archery puzzle

Maid Marian wants to try her hand at archery. She borrows Robin's bow and fires four arrows at the target. All are fairly wide of the bull's eye. Friar Tuck says "I don't think a fifth arrow has much of a chance of getting you closer to the bull's eye." What is the probability that it will?

Please develop your own answers to these questions before investigating the references below.  The class discussion of these problems appears at Probability paradoxes and simulation (Sibling Mystery, Three Coins, Envelope, and Monty Hall Problem). (see the "game show host" and "probability of boys" links) (archery) (airplane)

Do you have a favorite "paradox in probability" problem that you think should appear on this page?  Please send it to me (or e-mail a web reference).

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This page was created 31 January and last updated 23 April 2001.