Who doesn’t like a brief maths problem now and then?
Marcus du Sautoy, writing for The Times Online, placed this conundrum at the end of his article:
Let’s play a game. I’ll keep tossing a coin until one of the following two outcomes occurs. If heads, heads, tails, appears first then I’ll pay you £20. However, if it’s tails, heads, heads, then you pay me £10. Should you play?
No. It is three times more likely that tails, heads, heads appears. There are four possibilities for the opening two tosses: heads, heads; heads, tails; tails, heads; tails, tails. In the case of heads, heads, I can’t beat you. You just have to wait for a tail to appear. However, in the other three cases you can’t beat me. The first occurrence of heads, heads must be preceded by a tail, giving me the win.
Now on the face of it, this just seems wrong, as a number of commentators pointed out. There are 8 possible outcomes for three coin tosses, and you will win 1 in 8 and lose 1 in 8, so you should play the game (taking £20 each time you win and giving £10 each time you lose).
The problem, though, is not that Marcus is wrong, just that he doesn’t do a great job of explaining his result. For the sake of those not convinced by his explanation, here’s my attempt:
The idea is that we just keep tossing until one or the other sequence appears. Now we both need two heads in a row to win. However, you need your two heads to be followed immediately by a tail. All I need is that a tail immediately precede the two heads. Hence the only way you can achieve your sequence without me achieving mine first is if the first two tosses are heads (ruling out the possibility of them being preceded by a tails).
The other three combinations of pairs of starting tosses will result in my win because the first two heads that appear in a row will have to have been preceded by a tails (otherwise they won’t have been the first two heads in a row). So I win three out of four times.
Ok, so perhaps that’s just what he said but longer, but an interesting problem all the same. Of course, there are easier ways of taking money from people who are bad a maths, like the national lottery.