Say that one in every 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 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 .
Surprising? What if you were to then meet a further people? Then you’re quite a lot more likely to have seen an albino – 86% or – but still not guaranteed. Only after meeting 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.
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The probability of one randomly-assigned person being albino is , so the probability that they aren’t is .
Drawing 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 draws, the probability that none are albino is .
Thus, the likelihood of finding one or more is .
As increases, this converges quickly to a limiting value, which we find by taking the limit and taking the logarithm:
where we use the approximation . Hence .