Coin Chops: Can the Law of Averages be Replaced by the Law of Probability?

Alas, to the ‘average’ consumer of statistics, unlike we statisticians and data analysts, Probability is a sort of Comic I mean Cosmic Force i.e. ‘The Laws of Probability’ . David Hand OBE FBA has entertainingly looked at misunderstandings of this Comic Force and Coincidences in ‘the improbability principle: why coincidences, miracles and rare events happen all the time’ (2014).

But sitting here in the State Library of Victoria, I’m reading Frank ‘Power Without Glory’ Hardy’s novel ‘Four-legged lottery’ (1958). On page 179 of the Gold Star paperback edition there’s a bit of blarney about the ‘law of probability’ replacing the ‘law of averages’ where one of the two main characters, a professional gambler by the name of Jim Roberts, talks about the Anglo-Australian game of Two-Up which involves throws of pairs of coins, and is legal in Australian casino’s and traditionally, on the streets on ANZAC Day (25th April)..

‘in an honestly conducted two-up school, an equal number of heads and tails will be thrown over a long period; both head and tail bettor must lose [as the ‘house’ must take a percentage]. [To try and overcome the Law of Averages, giggle!] a tail better can back the tail on every spin – only for two throws, doubling [the] stake on the second throw if the spinner [bets heads] the first time. In this way [he or she] defeats the law of average <by winning> every time a spinner throws [both tails or one head and one tail and only loses when spinner throws both heads]’. Time for a simulation !

Watch this space. Same Stat Time! Same Stat Channel!


Who gives a toss: the statistics of coins

Spring is here in Melbourne, and a time for fashionable horse racing, including The Melbourne Cup in November., once attended by Mark Twain. Australia is also home of the “two-up” coin tossing game (descended from the British pitch and toss), played in outback pubs, hidden city lanes and now Australian casino’s, described in great old Australian novels such as Come In Spinner, and the eerie book and 1971 movie Wake in Fright (aka Outback).

In the 18th century, the Comte de Buffon obtained 2048 heads from 4040 tosses, while more recently and not to be outdone the statistician Karl Pearson obtained 12,012 heads out of 24,000 tosses (The Jungles of Randomness by Ivars Peterson, 1998). Of course a misunderstanding of the law of large numbers or so-called law of averages, makes the uninitiated think that if there’s say seven heads in a row, a cosmic force will decide “hang on that coin is coming up heads more than 50%, better make the next one a tail”).

While it doesn’t look at two-up, “Digital Dice” by the always entertaining Paul Nahin (2008) examines a tricky coin-tossing problem posed in 1941 and not solved until 1966. Prof Paul shows how to solve it using a computer-based Monte Carlo method, itself named after that famous casino in Monaco, where James Bond correctly observed that “the cards have no memory”.

And who says stats isn’t relevant?!

A Probability Book your Gran & Grandad could read: David Hand’s “The Improbability Principle”

Most people have heard of, or have actually experienced, ‘strange coincidences’, of the ‘losing wedding ring on honeymoon in coastal village and then years later, when fishing, finding the ring in the belly of a trout’ variety. Sometimes, the story is helped along a little over the years, such as the 1911 demise of Green, Berry and Hill who’d murdered Sir Edmund Berry Godfrey on *Greenberry* Hill, as used in the opening sequence of the 1999 Magnolia movie featuring the late great Philip Seymour Hoffman. The murder, however actually took place in the 17th century, and on *Primrose* Hill, which was later renamed to Greenberry Hill.

Still, odd things do happen, leading many to wonder ‘wow and what’s the probability of that!’. Strange events can however occur without the need for ghostly Theremin music to suddenly play in the background, in that they’re actually merely examples of coincidence, helped along by human foibles.

Coincidences and foibles are entertainingly and educationally examined in Professor David Hand’s excellent new 2014 book ‘The Improbability Principle: why coincidences, miracles and rare events happen every day’.

Prof Hand is an Emeritus Professor of Mathematics at Imperial College London, who like fellow British Statistician Brian ‘Chance Rules OK’ Everitt, has been writing instructive as well as readable texts and general books for nigh on forty years.

The book is not scarily mathematical at all, and illustrates using cards, dice, marbles in urns etc, although it might have been fun in the book, or at least the book’s website, to have some actual exercises that more active readers could have undertaken, using dice, cards or electronic versions thereof, such as the free Java version of Simon and Bruce’s classic Resampling Stats software, known as Stats 101 (commercial Excel version available at


All in all though, The Improbability Principle is not only highly readable, entertaining and inexpensive, it is an absolute snorter of a book, for a wide audience, including Uncles, Aunties, Grandmama’s and Grandpapa’s, and is thoroughly recommended!