Truly large numbers

Imagine an experiment which only works one time in a thousand. If you do that experiment a thousand times, what’s the probability that it works at least once? Counter-intuitively, it’s 63.2%.

Coin edge
allen watkin from London, UK / CC BY-SA

It’s a funny flaw in human reasoning that whenever we hear about something with very long odds – like a one-in-a-million chance – we think that means it’s impossible. It’s not, of course: million-to-one long shots happen every single day. In fact, even if something is extremely improbable, given enough time it is almost certain to occur at some point.

I’m going to drop a little math in here to illustrate. Consider a simple coin flipping experiment. Now most of the time when we flip a coin we want to see whether the result is heads or tails. If the coin is completely fair, there should be a 50% (0.5) chance of heads and 50% (0.5) chance of tails each flip. But I don’t care which side of the coin it lands on. I care whether the coin lands on either side, or the coin lands on its edge and stays that way. It’s incredibly unlikely, but it’s possible… so let’s give the edge a very low chance of occurring: one in a thousand.

The chance of a coin landing on its edge here is 0.001. The chance of it not landing on its edge, landing either heads or tails, is therefore 0.999. Pretty good odds. But here’s the trick: if I flip it twice, what is the chance of it not landing on its edge both times? Multiple the chance that the first flip is not an edge (0.999) by the same chance for the second flip (0.999) and you get 0.998001. Still extremely likely, but a tiny bit less likely than the first flip. Add a third flip and the probability of all three landing on a side rather than an edge drops again: 0.997003 (rounded up a little).

So what? Well, if we’ve worked out the chance of the coin always landing heads or tails, then the leftover part is the chance the coin lands on its edge at least once. With one flip, that chance is 0.001. With two flips, it increases to 0.001999. Three flips and it’s 0.0029997 (ish). That trend continues: every time we do another flip the chance that at some point the coin lands on its edge goes up. At a hundred flips, the chance of an edge is already up to 9.5%. At a thousand flips, the chance of an edge is at 63.2%. At ten thousand flips, the chance is up to a whopping 99.995% chance.

The higher you go, the more likely it is that the highly unlikely occurs. This is known as the law of truly large numbers – although it’s not really a law, more of an old saw. Yet another example of counter-intuitive but correct statistics.


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