One in a thousand

Say that one in every n people has some rare, special quality. (For argument’s sake I’ll use albinism – but it works for anything, visible or not, and it doesn’t even have to be people.) After you meet n people, what is the chance that you’ve met an albino?

You could be tempted to say it’s a near certainty. After all, if one in a thousand people are albinos, you expect to have met one after 1000 people, two after 2000 people, and so on…

But in truth, the answer isn’t 1: it’s around 63%, or 1 - 1/e.

Surprising? What if you were to then meet a further n people? Then you’re quite a lot more likely to have seen an albino – 86% or 1 - 1/e^2 – but still not guaranteed. Only after meeting 3n does the likelihood reach 95% – so you can safely say you’ve “probably” met one.

The maths behind this is below; it’s not hard, but not terribly gripping either. But it does (sort of) give the lie to a lot of casual remarks you might make, or hear in advertising campaigns.

For example, you may surf ten websites without ensuring yourself a malware infection (especially if you choose wisely.) And if you have three friends, it doesn’t have to be true that one of them will experience cancer.

But in a crowded room, it can still be sobering to choose your favourite statistic and map it on to your fellows.

– – –

The probability of one randomly-assigned person being albino is 1/n, so the probability that they aren’t is (1 - 1/n).

Drawing n people from a world population of 6 billion, we can assume the composition of the pool isn’t changed by each draw – and so we use unconditional probabilities. After an draws, the probability that none are albino is (1 - 1/n)^{an}.

Thus, the likelihood of finding one or more is P_{n,a} = 1 - (1 - 1/n)^{an}.

As n increases, this converges quickly to a limiting value, which we find by taking the limit n = \infty and taking the logarithm:

\ln (1 - P_{n,a}) = n \ln (1 - 1/an) \approx n [-1/n] = -a

where we use the approximation \ln(1+\delta) \approx \delta. Hence P_{n,a} = 1 - e^{-a}.

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