When Martin Gardner first presented the Two-Children problem, he made a mistake in its solution. Later he corrected the error, but unfortunately the incorrect solution is more widely known than his correction. In fact, a Tuesday-Child variation of this problem went viral in 2010, and the same flaw keeps reappearing in proposed solutions of that…

Quite a number of interesting problems in probability feature an event with probability equal to 1/e. This article discusses three such problems and attempts to explain why this probability occurs with such frequency.

If you break a stick at two random places, the probability that the three pieces form a triangle is 1/4. How does this generalize? To answer this question, we give a method for finding the probability that n randomly chosen points in a given interval fall within a specified distance of one another. We use this method to provide solutions to…

This article explores the question, "When should you mail in your entries to a sweepstakes in order to have the best chance of winning?"

The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of…

The "matching" hats problem is a classic exercise in probability: if "n" people throw their hats in a box, and then each person randomly draws one out again, what is the expected number of people who draw their own hat? This paper presents several extensions to this problem, with solutions that involve interesting tricks with iterated…

Consider the series [image omitted] where the value of each a[subscript n] is determined by the flip of a coin: heads on the "n"th toss will mean that a[subscript n] =1 and tails that a[subscript n] = -1. Assuming that the coin is "fair," what is the probability that this "harmonic-like" series converges? After a moment's thought, many people…

A discussion of the classic probability hat check problem. The paper shows how the hat check problem can be viewed as a problem about enumerating certain members of the symmetric group of permutations on n symbols, with an ensuing generalization. (PK)