During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed on the web, with much misunderstanding and confusion. I will try to avoid adding to that.

## Posts Tagged 'Probability'

### The Two Envelopes Fallacy

Published November 29, 2018 Occasional 1 CommentTags: Probability, Puzzles

### The Improbability Principle

Published April 6, 2017 Irish Times Leave a CommentTags: Probability, Statistics

*Extremely improbable events are commonplace.*

“It’s an unusual day if nothing unusual happens”. This aphorism encapsulates a characteristic pattern of events called the *Improbability Principle*. Popularised by statistician Sir David Hand, emeritus professor at Imperial College London, it codifies the paradoxical idea that extremely improbable events happen frequently. [TM112 or search for “thatsmaths” at irishtimes.com].

### Treize: A Card-Matching Puzzle

Published March 30, 2017 Occasional Leave a CommentTags: Probability

Probability theory is full of surprises. Possibly the best-known paradoxical results are the Monty Hall Problem and the two-envelope problem, but there are many others. Here we consider a simple problem using playing cards, first analysed by Pierre Raymond de Montmort (1678–1719).

### Twenty Heads in Succession: How Long will we Wait?

Published December 22, 2016 Occasional Leave a CommentTags: Probability, Statistics

If three flips of a coin produce three heads, there is no surprise. But if 20 successive heads show up, you should be suspicious: the chances of this are less than one in a a million, so it is more likely than not that the coin is unbalanced.

Continue reading ‘Twenty Heads in Succession: How Long will we Wait?’

### Random Harmonic Series

Published July 28, 2016 Occasional Leave a CommentTags: Analysis, Number Theory, Probability

We consider the convergence of the random harmonic series

where is chosen randomly with probability of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

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.

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?*

### Franc-carreau or Fair-square

Published February 11, 2016 Occasional Leave a CommentTags: Games, Geometry, History, Probability

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?*