This page is devoted to solved problems that can be simply stated but which require special insight or a special mathematical technique. Less weighty problems of this sort can be found on the Marilyn Is Right web site or on the Jersey Devil Puzzle Pages.

The Monty Hall Problem

This is the problem most notorious for generating wrong answers. It concerns the strategy that affords the best chance of winning a prize on a television game show. It is intimately connected with the name of Marilyn Vos Savant and our discussion is to be found on the Marilyn Is Right web site.

Eddington's Liars Problem

This is a problem not intended to be solved but a few people have solved it anyway. Discussion

The Biased Coin

A biased coin has the probability p of coming up heads and the probability q of coming up tails. If it is tossed indefinitely, what is the probability that at some time or other you will get more tails than heads? What is the probability of ever getting an equal number of tails and heads? Discussion

The Pay Envelopes

This problem seems to trigger a glitch in human logic in the same way that The Monty Hall Problem does. The paymaster puts two envelopes labeled A and B on the table and invites you to choose one of them. You take envelope A and find that it contains $50. The paymaster says "One of these envelopes contains twice as much money as the other. Maybe you chose the wrong one. There is a 50% chance that envelope B contains $100 and a 50% chance that it contains $25, so your expected value of the money in envelope B is $75. If you pay me $1 I'll let you have envelope B instead." Should you pay him the dollar and switch envelopes? He would have said the same thing if you had chosen envelope B.

A great may explanations have been offered for this paradox, and a summary may be found here. Here is ours. Given that the envelopes contain S and 2S dollars, the expected value of each envelope is 3S/2 while both are still on the table. However when you remove one envelope from the table the conclusions made about the population of two envelopes are not necessarily valid when applied to a population of one envelope. If you compute the statistics of of population of data items and then remove some of the items, in general the remaining ones will not produce the same statistics as the original population.

The Birthday Paradox

Discussion

The Multiple Wins Paradox

Discussion

The Secretary Problem

An interviewer may see a number of job applicants whose qualifications can be given a numerical measure. He must make an immediate decision whether or not to hire any particular applicant. What strategy offers the best chance of hiring the best qualified applicant? This problem sometimes is known as "The Sultan's Dowry Problem" in which a suitor tries to choose the sultan's daughter with the largest dowry. It originated in Martin Gardner's "Mathematical Recreations" column in Scientific American and has been worked on by mathematicians of the first rank. Papers covering variations of it continue to appear. Discussion

The Gossip Problem

A community consists of m people, each of whom knows something special. What is the best strategy for transmitting all of the information to everyone by telephone and how many telephone calls will be required? Conference calls are not permitted. Discussion

The Crackerjack Box Problem

In each Crackerjack box, in addition to the crackerjack, there is a charm. If there are 36 individual charm designs and each box is equally likely to contain any of them, how many boxes must I buy to be 90% sure of getting a complete set of charms? Discussion

Probability of a Run

A gambler who uses a Martingale or "doubling up" betting strategy will be ruined by a long enough string of successive losses. Where an event has a probability p, what is the probability of a run of at least k consecutive occurrences in n trials, u(k,n)? This is an old problem but an exact solution remains elusive to this day. Discussion