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 [2012]. We will have a look at Bertrand’s Paradox (1889), a simple result in geometric probability.

Bertrand-00Let’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?

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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.

Kangaroo-Frog-01

Black: Frog hop. Blue: Cricket hops. Magenta: Flea hops.

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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.

Franc-Carreau-01

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?

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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

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].

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