Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?

Continue reading ‘Think of a Number: What are the Odds that it is Even?’

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Tags: Number Theory, Probability

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?

Continue reading ‘Think of a Number: What are the Odds that it is Even?’

Tags: Probability, Statistics

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].

Continue reading ‘How many numbers begin with a 1? More than 30%!’

Tags: Probability, Puzzles

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.

Tags: 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].

Tags: 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).

Tags: 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?’

Tags: 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?*

Tags: 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?*

Tags: Analysis, Probability

Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.

The Buffon Needle method of estimating is hopelessly inefficient. With one million throws of the needle we might expect to get an approximation accurate to about three digits. The idea is more of philosophical than of practical interest. Buffon never envisaged it as a means of computing .

Continue reading ‘Buffon was no Buffoon’If we toss a `fair’ coin, one for which heads and tails are equally likely, a large number of times, we expect approximately equal numbers of heads and tails. But what is `approximate’ here? How large a deviation from equal values might raise suspicion that the coin is biased? Surely, 12 heads and 8 tails in 20 tosses would not raise any eyebrows; but 18 heads and 2 tails might.

Tags: Ireland, Probability, Statistics

This week, *That’s Maths* in *The Irish Times* ( TM044 ) is about the originator of Students t-distribution.

In October 2012 a plaque was unveiled at St Patrick’s National School, Blackrock, to commemorate William Sealy Gosset, who had lived nearby for 22 years. Sir Ronald Fisher, a giant among statisticians, called Gosset “The Faraday of Statistics”, recognising his ability to grasp general principles and apply them to problems of practical significance.

Tags: Geophysics, Number Theory, Probability

In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first *N* terms. For geometric series, like 3 + 6 + 12 + 24 + … , each term is a fixed multiple of the previous one. Again, there is a formula for the sum of the first *N* terms of such a series. Continue reading ‘Breaking Weather Records’

This week, *That’s Maths* ( TM018 ) deals with the “war” between Bayesians and frequentists, a long-running conflict that has now subsided. It is 250 years since the presentation of Bayes’ results to the Royal Society in 1763.

The column below was inspired by a book, *The Theory that would not Die*, by Sharon Bertsch McGrayne, published by Yale University Press in 2011.

Continue reading ‘Bayes Rules OK’