How to win a coin toss 51% of the time

Flipping a coin isnt fair

A natural bias is increasing the odds of turning up heads when tossing a coin.

As the proud captain of the Kent Over 60s cricket team (OK, the third XI), I was beginning to wonder how it has come about that I have won all four tosses this year. True, I did lose it in the pre-season friendly, but then I hadn’t practised this vital part of the job all winter. I have always worked to the maxim of “Heads in Kent, Tails Elsewhere” and it works for me!

You don’t need to be a mathematician or a Vegas card shark to know that, when all things are equal, the probability of flipping a coin and guessing which side lands up correctly is 50-50. (Note that I can flip a coin and get the outcome I want every time – let’s just say that I have to catch it and that’s all I’ll let on.)

But seriously, it seems that the toss of a coin is not a 50-50 chance after all. Not if Stanford University’s Professor of Mathematics and Statistics Persi Diaconis is right. According to Diaconis, named two years ago as one of the “20 Most Influential Scientists Alive Today”, a natural bias occurs when coins are flipped, which results in the side that was originally facing up returning to that same position 51 per cent of the time.

In late March this year, Diaconis gave the Harald Bohr Lecture to the Department of Mathematical Sciences at the University of Copenhagen, but focused mainly on his famed reputation for predicting the randomness of card shuffles, with the theme “Adding Numbers and Shuffling Cards”. A magician in his spare time, Diaconis was famous for discovering that it requires either five of seven shuffles, depending on the criteria, to get a deck of cards into a mathematically random order.

But Diaconis has been publishing papers (and latterly videos and blogs) on coin tossing since 1978. Expressed in simple terms, Diaconis came to this conclusion after determining that no matter how hard a coin is flipped, most of the time, the side that started face-up will spend more time facing up. A similar result is obtained by looking at the ratio of even and odd numbers starting from one: no matter what number you stop at, there will never be more even numbers than odd numbers in that sequence. And this ultimately leads to a small bias in the attitude of the coin when it comes to rest.

Looked at another way, the secret of gaining this slight advantage is to make sure you carefully observe the starting position of the coin in your opponent’s hand!

Perhaps more remarkably, although less difficult to explain, is that spinning a coin gives much more pronounced oidds than this. This is because the centre of gravity of the coin lies towards the heavier (heads) side and this causes the coin to lean towards the head and thus favour landing tails side up – in some cases, as much as 80:20.

In this month of mathematical consternation, as the country’s youth grappled tentively with a GCSE question about orange and yellow sweets in a bag (there were 6 orange ones and 4 yellow ones by the way), we wonder how thick would a coin need to be such that there was a 1/3 chance of heads, 1/3 chance of tails and 1/3 chance of it landing on its edge?

Andy Pye
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