The gambler’s fallacy

‘Dunfermline are due a win at Tannadice.’
Dunfermline defender Scott Wilson, Daily Record, 30 October 2004

When people say they are “due a win”, in sport, gambling or, more metaphorically, in life in general, they are more often than not doing little more than expressing a hope born of despair. But sometimes they also believe that in a very literal sense their luck is due to change.

The idea, usually vaguely rather than explicitly held, is that nature balances things up in the long run, so a recent run of results going one way requires a balancing set of results going the other. Otherwise, as Hamlet might put it, the world is out of joint.

Perhaps the clearest evidence that many people do think like this is the popularity of web sites that tell you how many times numbers have been picked in national and state lotteries. (You can find one such “resource” here.) The fact that people consult these sites only makes sense if they believe that past selections can provide some kind of indicator of the likelihood of future ones.

The futility of this kind of analysis is evident from the fact that you could interpret the past data in two totally opposite ways. On one view, the numbers that come up most frequently are luckier, and so should be selected. On the other, the ones that have come up least frequently are overdue for an appearance. So which do you pick?

The answer is neither. The former is mere superstition, and the latter is a muddle-headed way of thinking about random events which falls prey to what is known as the gambler’s fallacy. The fallacy rests on a misapprehension about the probability of random occurrences. Take the toss of a fair coin. The chances of it coming up heads or tails is 50/50. That means that if you toss it 100 times, it is likely to come up heads about as often as it will come out tails. But note it is more likely that the outcome will be an unequal amount of heads and tails than exactly 50 heads and 50 tails. That’s because each toss of the coin is a distinct event and does not affect the tosses that follow or precede it. So if there have been 49 heads and 50 tails, nature does not “know” that a head is due. Rather, that 100 th toss, like the other 99 that preceded it, is as likely to come up heads as tails.

On that analysis, it is a misunderstanding of the nature of probability as it relates to statistical averages over a series of events that is the problem. Another source of the mistake is to misapprehend the nature of unlikely events. For example, the chances of tossing ten heads in a row is one in 1,024. (My maths may actually be wrong here, but it doesn’t matter as the general point being made is, I am sure, right!) Let us say that we have tossed a coin nine times in a row and it has come up heads every time. So surely, people feel, since the chances of a series of ten is so unlikely, it must be more likely that the next toss is tales rather than heads? Wrong. The unlikely (1 in 512, I believe) sequence of nine heads prior to this toss does not affect the outcome of the tenth toss. What is really unlikely has already happened. The probability that a tenth head will be tossed from this starting point is thus not 1 in 1,024 but 50-50, because it all hinges on one toss which, like all the others, is an evens bet.

Although it is a straightforward fallacy to suppose past random events affect the outcome of future ones, you might nonetheless justifiably see an unlikely series as being evidence that the sequence isn’t random at all. For example, if a coin has been tossed heads nine times in a row, you might bet on heads for the tenth on the basis that you suspect the tossing isn’t random. That’s perfectly reasonable, just as long as you understand that the mere occurrence of the unlikely series does not in itself show that the tossing has not been fair. It is of the nature of unlikely events that they will occasionally happen.

What isn’t reasonable is to believe that previous coin tosses, spins of the roulette wheel, or pieces of misfortune in competitive sport add or subtract something to the probability that the same chance events will or not occur the next time. And that’s why neither luck nor natural forces will ever make you due a break just because you’ve been on a losing streak. Sorry, Dunfermline football club.

Note: this is the 50th Bad Move. Many thanks to everyone who has followed the series and commented on it. I am planning at least 50 more before the series draws to an end.

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