Marcus: Thomas and I are going to play a game called Penney’s game, which isn’t named after the penny that we will be tossing but after Walter Penney, who came up with this game in 1969.
The way you play this game is you each make a choice of three tosses, either HHT or HHH, and the person who wins is the one whose combination comes up first when we toss a coin again and again and again.
So Thomas, I want you to make a choice of three tosses of the coin that you think will come up.
Thomas: Okay, so it is not a biased coin in any way.
Marcus: No, 50/50, heads or tails.
Thomas: Okay, so 50/50, heads or tails equally likely. I might as well go for HHT.
Marcus: Now the point about this game is that if you know your mathematics, you can get an edge in playing this. I’m going to make a choice now, which applies a little mathematical thought that is going to give me an edge when playing against Thomas. So applying a little algorithm I am using, I am going to choose THH.
Now I’m going to start tossing the coin and will keep on tossing it until one of these sequences appears and that will be the winner.
[Coin is tossed and results are: HTTHH]
Marcus: So my strategy did work. It’s not guaranteed, this is a probabilistic game, but it does certainly give me an edge. On average, the strategy that I am using will mean that I will win many more times than Thomas will.
You might have noticed there that the only way that Thomas could have won would have been if there was a H to start. But as soon as there was a T, Thomas was doomed and my strategy kicked in. Thomas needed HH and provided there is a T before that, I would win.
What we want you to do is to analyse this game and come up with the strategy that gives me the edge over Thomas.