Long shots

March 5, 2007

Ever wanted to see a one in a million chance event actually happen? Probability is one of those areas of maths that typical members of the public have a basic conceptual grasp of, but generally a pretty poor detailed understanding of. This is why things like the National Lottery can exist and make money.

I think in general people don't have an intuitive feel for what one in a million means. They don't really get just how long a shot it is. So, try this little experiment: Get a coin and flip it twenty times, and write down H for each heads, and T for each tales. Keep the H's and T's in order. You should end up with something like:


I used this rather obscure web page to speed up my coin flipping. Now, unsurprisingly I didn't get all coins come in heads. You didn't too, I'd be willing to bet. If you did, you've seen your one in a million event and can stop reading now, as the chance of getting heads from each of 20 coin flips is 1 in 1,048,576. That's one in a million, roughly.

I'd also be willing to bet you didn't get the sequence of heads and tails that I got above. I'll also wager that if you keep playing, you won't get that exact sequence before you get bored trying. The thing is that every sequence* of 20 coin flips has the same probability - about one in a million.

So there you go. Your one in a million event has happened. There was a one in a million chance of the particular sequence of results you saw, and you still saw it.

Ok, so that should give you an idea of what one in a million means. Now lets add some more players to the game. Lots more. Assume this article is being read by ten million people (I know - even my ego doesn't imagine that to be true). Ten million people all run this experiment, flipping a coin 20 times each. It seems hard to imagine, but about ten of those people will actually get a result of all heads. About ten more of them will get my exact sequence from above. Another ten or so will get the sequence you got. How lucky would you feel to get all 20 heads? You shouldn't feel any luckier than if you'd got any other sequence. They're all equally likely.

* Note that a sequence of coin flips is a very different thing to counting up the number of heads and tails in a set of flips. Ie, in my example run I got a pretty typical 11 heads and 9 tails. You will be able to replicate that in just a few experimental runs of course. It is the sequence of the results that are important in this instance.


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  1. Ciz says:
    March 5, 2007 @ 21:30 — Reply

    Completely wrong, I'm afraid. It's easy to see that on *average* the probability of getting a typical sequence of 20 heads-or-tails is 1/(2^20). But, when you toss five heads in a row, then on the sixth toss you're virtually *guaranteed* to get a tail because you haven't had one for so long! You might not understand so let me give you a more real-world example. If you play the lottery (which you should, because it's a great way to make money, although I'm still waiting for my big win, I calculate it should be any week now) then if your favourite numbers haven't been drawn for a few weeks then, again, they're almost *guaranteed* to be drawn the next week. If you're clever, you can look at lottery ball frequency statistics to work out which numbers will come up next. That's just one tip you can have for free, there are plenty of books out there with other brilliant tips on how to win the lottery. It's a great way to make money, and yet another example of how the real world differs from the opinions of mathematicians.

  2. Vaughany says:
    March 6, 2007 @ 16:30 — Reply

    What Ciz said, with the addition of: "The national lottery is a tax on people who are bad at maths."

  3. Rockvole says:
    March 7, 2007 @ 14:26 — Reply

    You think the odds of winning the lottery are low? Thats nothing when compared to the chances of the unlikely series of events in every episode of prison break happening. I mean each week it seems like there is a million to 1 chance that those unlikely events could happen in that order without Michael being caught or killed.
    I guess if there is an infinite number of universes with every combination of events happening it could really happen, but really I think you would need more than infinity for a whole series of prison break. To me, most episodes now are about as bad as that moment in Dallas when Bobby Ewing stepped out of the shower after he had been killed off in the series 18 months previously.

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