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!


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!


Spiders, sowbugs and sundew statistics

Statisticians often like to think that non-statisticians don’t know what exactly it is that we do. The truth is of course that not only do they not know, they do not particularly care! With the possible exception of someone like Nat ‘2012 US election’ Silver, what statisticians are thought to do is about as exciting as driving around in  cardigan and slippers in a two-tone ’74 Morris Marina with no radio.

But what  if statisticians went around ripping up floorboards and counting up spiders? Now you’re talking!!

Back in ’46 a scientist named LC Cole published some data on counts of spiders, and sowbugs, (or woodlice, roly poly’s or slaters).

Cole and various bright sparks ever since, had the idea of fitting the spider / sowbug counts to various types of probability distribution. Voila!, it was found that spider counts could be quite happily fitted by the Poisson distribution, as can the number of typewriter errors made on a page, the number of people killed by horse kicks in the Prussian cavalry, etc etc.

But not sowbug counts, which are better fitted by a ‘contagious distribution’, such as the ‘generalized Poisson’ or ‘generalized Negative binomial’, in which the event of something happening is itself dependent on other events. Sowbugs, it seems are a social breed, and when they notice their numbers dwindling, to the point where there’s only one or two left, they pick up sticks and try the house down the road, in search of other sowbugs, if not adventure.

Spiders, on the other hand, are more individualistic or anti-social and don’t care if they’re left by themselves.  (In fact they probably appreciate the peace and quiet after those pesky sowbugs have marched off elsewhere, unless of course the spiders belong to the  type known as woodlouse spiders or sowbug hunters, which is a very different kettle of fish, or spiders, altogether, as are ‘shy spiders’and ‘social spiders’)

Finally, a paper published in the journal of the highly prestigious Royal Society in 2010 found that carnivorous wolf spiders (Lycosidae) and pink sundew plants (Drosera capillaris) competed with each other for available food, in statistically interesting ways, indeed the lead author described the study as ‘awfully fun’

So, next time someone asks (without really caring what the answer is) ‘just what is it that statisticians actually do.?….’.

[updated, 9 October 2016]


Cole LC (1946) A study of the cryptozoa of an Illinois woodland. Ecological Monographs, 16, 49-86.

Consul PC (1989) Generalized Poisson distributions. Marcel Dekker, New York.

Forbes C, Evans M et al (2011) Statistical distributions. 4th ed. Wiley, Hoboken, New Jersey.

Janardan KG et al. (1979). Biological applications of the Lagrangian Poisson distribution. BioScience, 29, 599-602.
Jennings DE et al. (2010). Evidence for competition between carnivorous plants and spiders. Proc Royal Society B, 277, 3001-3008.

Raja TA, Mir AH (2011). On applications of some probability distributions. Journal of Research & Development, 11, 107-116.