Posts Tagged 'Probability'

How many numbers begin with a 1? More than 30%!

The irregular distribution of the first digits of numbers in data-bases provides a valuable tool for fraud detection. A remarkable rule that applies to many datasets was accidentally discovered by an American physicist, Frank Benford, who described his discovery in a 1938 paper, “The Law of Anomalous Numbers” [TM181 or search for “thatsmaths” at irishtimes.com].

Benford-Distribution-3

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The Two Envelopes Fallacy

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.

Two-Envelopes

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The Improbability Principle

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].

Improbability-Principle-Top

From front cover of  The Improbability Principle

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Treize: A Card-Matching Puzzle

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).

SpadesHearts

Shuffle spades in one pile, hearts in another. Place both piles face downwards. Turn over a card from each pile. Do the two cards match?

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Twenty Heads in Succession: How Long will we Wait?

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.

euro-frontback-ie

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Random Harmonic Series

We consider the convergence of the random harmonic series

\displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n}

where {\sigma_n\in\{-1,+1\}} is chosen randomly with probability {1/2} of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

RandomHarmonicSeriesDistribution

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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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