We find the probability of winning a best-of-three racquetball match given the probabilities that each player wins a point while serving.

We construct semigroups with any given positive rational commuting probability, extending a Classroom Capsule from November 2008 in this Journal.

We consider the expected range of a random sample of points chosen from the interval [0, 1] according to some probability distribution. We then use the notion of convexity to derive an upper bound for this expected range which is valid for all possible choices of this distribution. Finally we show that there is only one distribution for which this…

This paper surveys the fascinating mathematics of fair division, and provides a suite of examples using basic ideas from algebra, calculus, and probability which can be used to examine and test new and sometimes complex mathematical theories and claims involving fair division. Conversely, the classical cut-and-choose and moving-knife algorithms…

The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite…

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…

A standard mattress can be positioned on a bed frame in any of four orientations. Suppose that four times a year the mattress is rotated into one of the three possible new orientations, chosen at random. According to an article in The American Scientist, a computer simulation suggests the rather surprising result that over a period of 10 years,…

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…

The author describes a card trick that failed when he tried it with the student chapter at his university. Computations show that the chance of this happening is about 1 in 25.

A result known as the Borel-Cantelli lemma is about probabilities of sequences of events. This article presents an example in which it appears that the hypotheses of the lemma are satisfied but the conclusion is not. The explanation of why not combines elements of probability theory, number theory, and analysis.

Within the set of points in the plane with integer coordinates, one point is said to be visible from another if no other point in the set lies between them. This study of visibility draws in topics from a wide variety of mathematical areas, including geometry, number theory, probability, and combinatorics.

Developed are simple recursion formulas for generating the exact distribution of the longest run of heads, both for a fair coin and for a biased coin. Discusses the applications of runs-related phenomena such as molecular biology, Markov chains, geometric variables, and random variables. (YP)