IMAGINE you receive an envelope addressed in an unfamiliar hand. Enclosed are predictions for this weekend’s football matches and an offer to invest in the sender’s foolproof betting syndicate. What tosh, you think, shoving it in the recycling bin.
But come the weekend, you notice that those tips turned out to be correct. And then comes the really strange bit. The next week, an identical letter arrives with predictions for that weekend’s games – and they turn out to be accurate too.
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At this point, you send off your cash, convinced that whoever this person is possesses some genuine insight. (Either that, or you go to the police to report that you’ve uncovered the biggest match-fixing scandal yet.)
Or if you’re familiar with the law of large numbers, you might be tempted to bide your time. This law, a facet of the perennially bamboozling subject of probability, states that, given a large enough sample size, any outrageously improbable thing is eventually bound to occur. If our sly soothsayer simply sent letters systematically to enough different people, each with a different set of scores, then at least one recipient is likely to get accurate predictions enough times in a row to make them bite. And even if just a few people hand over their money, it probably makes the scam worthwhile.
“Given a large enough sample, any improbable thing is eventually bound to occur”
This sort of trick works so well because the existence of these hundreds of disappointed punters never occurs to us. “It’s very difficult to count all the times something could have happened and didn’t,” says David Spiegelhalter, a statistician at the University of Cambridge.
Another way this plays out is in the birthday paradox. You’re at a party with 23 guests and are surprised and delighted to find that two guests share a birthday – what a coincidence! In fact, a little calculation shows you the odds of this happening are better than 50:50. Again, the crucial point is the number of possibilities. One person in the room can share a birthday with any of the 22 others, and there are 365 days on offer in a standard year for this to happen. A second person can also share a birthday with any of the others, and so on. Continue adding up the possibilities for each person, and you end up with enough to make a shared birthday more likely than not.
We have this blind spot for large numbers for good reason. Being hypersensitive to “coincidences” was handy in our evolutionary past. “By following coincidences we make important discoveries,” says Tom Griffiths, a cognitive scientist at the University of California, Berkeley. We first had to realise that the sun rises every day, and search for an underlying cause, before we could (eventually) conclude that Earth rotates. Equally the recurrence of stomach pains and worse taught us to stay away from certain berries.
“We just happen to live in a world where most of the relevant causal relationships have already been discovered,” says Griffiths. The result is we see patterns where none exist. Our problem with coincidences today, then, is a relic from a simpler world, like cognitive bias (see “Cognitive bias“).
Looking past what our gut tells us, at the often hard truths revealed by probability, calculations can also help us in balancing the risks we all face in our daily lives, from not taking an umbrella to not taking out an insurance policy. That probability is so useful is no coincidence.
Like this? Read: “Get smarter: 9 ideas to make your life better“
(Image: Bruno Mangyoku)
This article appeared in print under the headline “Need to know: Probability”
Article amended on 8 January 2016
Correction: Since this article was first published, it has been updated to reflect that in a group as small as 23 people, one shared birthday is more likely than none.
