I was recently invited by Talksport Radio to be their “maths expert” to speak on the topic of coin tossing, in the context of determining who bats first in a cricket match. Here’s a fuller version of what I said.
One of the best-known studies of coin tossing (at least amongst statisticians) was carried out by John Kerrich. He was interned by the Nazis in Belgium during World War II. Being a mathematics lecturer he took the opportunity to carry out a series of experiments in probability, including tossing a coin 10,000 times.
He noted two interesting phenomena.
First, the difference between the number of heads you expect to get and the number of heads you actually get tends to increase as the number of coin tosses increases. In his first hundred tosses, he got 44 heads, which is 6 fewer than you would expect. After ten thousand tosses, he had got 5,067 heads, which is 67 more than you would expect, more than ten times as many.
Second, although the absolute difference increases, the relative difference decreases. In other words, the proportion of heads tossed becomes closer to what you expect as the number of tosses increases. After the first hundred tosses, he’d had 44% heads, which is 6% below what you’d expect. But after the full ten thousand tosses, he’d had 50.67% heads, which is only 0.67% above what you’d expect.
The graph above, which is based on Kerrich’s data, explains why casinos are successful businesses. Consider someone playing blackjack in a casino. The odds of winning any particular hand are roughly 49:50, so it’s almost like tossing a coin. A typical player might play a hundred hands. So he’ll be operating on the left-hand side of the graph. Here wild, unpredictable swings are very typical. This is exciting: you might lose a lot and you might win a lot.
But the casino is playing against many different players and pretty much 24 hours a day. So they are playing thousands of hands and are thus operating on the right-hand side of the graph. Here the trend has clearly established itself. There is little variation: despite the random nature of any given hand, over thousands of hands the pattern is clear – steady, virtually guaranteed winnings for the casino.
So in the short-term, the outcome of a coin toss is inherently unpredictable, but in the long-run the overall outcome is very predictable.
This begs the question: can you consistently toss heads rather than tails? An experiment carried out in 2009 set out to test this. Thirteen otolaryngologists (doctors who specialise in the ear, nose and throat) were asked to toss a coin 300 times each, with the intention of tossing as many heads as possible. All of them tossed more heads than tails: the least being 160 heads (10 more than expected) and the greatest being 203 heads (53 more than expected).
There is of a course a natural variation in the number of heads you would actually get if you toss a coin 300 times. You wouldn’t expect exactly 150 heads every single time. But even taking this into account, five of the 13 doctors got results that you would expect to see less than one time in a hundred. Pretty compelling evidence that you can bias the outcome of a coin toss in the way you toss the coin.
Another way of achieving this would be to use a biased coin: one which is weighted so as to give more heads than tails. There was much excitement in 2002 when a group of Polish statisticians claimed that the Belgian one-euro coin was biased in exactly this way. They claimed that in 250 spins (they preferred spinning the coin rather than tossing it) heads came up 140 times, which is 56% of the time. They believed the heavy embossed image of the King of Belgium on the heads side of the coin was responsible for the bias. Sadly, however, a result such as 140 heads of 250 is not particularly unlikely for a perfectly fair coin. This result was not as significant as that achieved by the otolaryngologists.
Finally, is there a way to predict the outcome of a coin toss using physics? It has certainly been shown that you can use physics to predict the outcome of a spin on a roulette wheel, by taking measurements of the ball’s speed. The physics of a coin toss is, I suspect, much more complicated: it has more degrees of freedom in its movement, compared with a roulette ball which is constrained to run around its fixed track. I’m reminded of the scene in Jurassic Park where Jeff Goldblum’s character explains chaos theory. He dribbles some water down Laura Dern’s hand. Each time he does it, minute variations in her skin cause the water to follow a slightly different path. The same effect may well occur as a coin spins through the air.