The Law of Averages

If you toss a "fair" coin, then in the long run heads and tails should even out. Colloquially, this fact is known as the "law of averages" and it is often misapplied, which is why statisticians acknowledge no law by that name. However, the law of large numbers provides a sense in which it is true.

A coin has no memory: future tosses happen without any input from past ones. How, then, does the coin "know" that the numbers should even out? Many people think that if a coin has come up heads a lot more times than tails, then tails becomes more likely. (The worldly wise think the opposite: the coin is probably biased towards heads.) But fair coins can perfectly easily produce a run of heads. I once threw 17 consecutive heads with a normal coin, an event with probability 1 in 131,072.

So you've got a fair coin but you have just thrown 17 heads. How does the number of tails catch up to make the proportions equal? Surely tails must now become more likely? Not so. The next toss is just as likely to produce another head as a tail, and the same goes for all subsequent tosses. In the long run, subsequent tosses should be very close to half heads, half tails. So, in 2 million additional tosses, we expect, on average, a million heads and a million tails.

Although 17 is very different from 0, 1,000,017 is proportionately much closer to a million: their ratio is 1.000017, very close to 1. Instead of tails catching up with heads, the future tosses swamp the first few, and the longer you keep tossing the less important that initial difference becomes.

Newspapers publish lists of how frequently numbers appear in the UK's Lotto draws. At one stage 13 was relatively infrequent, reinforcing the view that 13 is unlucky. Some people therefore expect 13 to come up more often in future. Others think that its unluckiness will persist. The mathematics of probability, and betting, supported by innumerable experiments, says both camps are wrong.

In future, all numbers have the same chance of being picked. The lottery machine treats all balls alike, and it doesn't "know" what number is written on them.

Paradoxically, that does not mean every number will turn up equally often. Exact equality is highly unlikely. Instead, we expect to see fluctuations about the average value, with some winners and some losers. The mathematics even predicts the size and likelihood of those fluctuations. What the maths can't do is predict which numbers will be winners and which losers. In advance, it could equally well be any of them.

Ignore the newspaper tables: they belong to a dead past and tell you nothing about what will happen in next week's draw.