## Archive for February, 2016

### Bertrand’s Chord Problem

The history of probability theory has been influenced strongly by paradoxes, results that seem to defy intuition. Many of these have been reviewed in a recent book by Prakash Gorroochurn . We will have a look at Bertrand’s Paradox (1889), a simple result in geometric probability. Let’s start with an equilateral triangle and add an inscribed circle and a circumscribed circle. It is a simple geometric result that the radius of the outer circle is twice that of the inner one. Bertrand’s problem may be stated thus:

Problem: Given a circle, a chord is drawn at random. What is the probability that the chord length is greater than the side of an equilateral triangle inscribed in the circle?

### Vanishing Zigzags of Unbounded Length

We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity. Black: Frog hop. Blue: Cricket hops. Magenta: Flea hops.

### Franc-carreau or Fair-square

Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player’s coin lands completely within a square, he or she wins a coin of equal value. If the coin crosses a dividing line, it is lost. The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.

The question is: What size should the coin be to ensure a 50% chance of winning?

### The Mathematics of Voting

Selection of leaders by voting has a history reaching back to the Athenian democracy. Elections are essentially arithmetical exercises, but they involve more than simple counting, and have some subtle mathematical aspects [TM085, or search for “thatsmaths” at irishtimes.com]. Rock-paper-scissors, a zero-sum game. There is a cyclic relationship: rock beats paper, paper beats scissors and scissors beats rock [Image: Wikimedia Commons].