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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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Numerical Coincidences

A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.

Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!

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A Life-saving Whirligig

Modern science is big: the gravitational wave detector (LIGO) cost over a billion dollars, and the large hadron collider (LHC) in Geneva took decades to build and cost almost five billion euros. It may seem that scientific advances require enormous financial investment. So, it is refreshing to read in Nature Biomedical Engineering (Vol 1, Article 9) about the development of an ultra-cheap centrifuge that costs only a few cents to manufacture [TM111 or search for “thatsmaths” at irishtimes.com].

SAMSUNG

Whirligig, made from a plastic disk and handles and some string

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Brun’s Constant and the Pentium Bug

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:

\displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty

Obviously, this could not happen if there were only finitely many primes.

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Enigmas of Infinity

Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com]. 

infinity-symbols

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Topology in the Oval Office

Imagine a room – the Oval Office for example – that has three electrical appliances:

•  An air-conditioner ( a ) with an American plug socket ( A ),

•  A boiler ( b ) with a British plug socket ( B ),

•  A coffee-maker ( c ) with a Continental plug socket ( C ).

The problem is to connect each appliance to the correct socket, avoiding any crossings of the connecting wires.

electricplugs-01

Fig. 1: Positions of appliances and sockets for Problem 1.

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