Penney’s Game
Here is a trick you can do with pennies. It is called Penney’s game. It is named after Walter Penney. Or was he named after the game? No one knows.
You choose a sequence of heads and tails, three in total. Say HHH. I then choose another sequence, say THH. We now toss a coin over and over, noting the order of heads and tails. If your sequence occurs first then you win, if mine then I win.
Now surely is obvious that both players have an equal probability of winning? Simply because heads and tails are equally likely, as are all the combinations of the three in the sequence. But this not obvious to the mathematician. And here the mathematician is right.
(If you thought equally likely then I’ve got a bridge to sell you.)
Mathematicians never take anything for granted. Even the most obvious idea must be rigorously proved before the mathematician can get any sleep. Here’s something that surely is obvious, but turns out not to be. And it makes a great trick to play in a bar. Yes, another one. Some people say I spend too much time in bars. I say, “Your round.”
With the two sequences chosen here, the first player has HHH and the second THH, the second player has a seven in eight chance of winning. Now how can that be?
We can see this easily with the above choices for the two players. If ever a T is tossed then the first player cannot possibly win:
XXXXXTHH
has to come before
XXXXXTHHH.
Therefore the only way the first player can win is if the first three tosses are all heads, which has a small one in eight chance. The same principle applies to other combinations, albeit not so trivial to demonstrate. In this table we see how the second player should choose his sequence to maximize his probability of winning:
| 1st player’s choice | 2nd player’s choice | Odds in favour of 2nd player |
| HHH | THH | 7 to 1 |
| HHT | THH | 3 to 1 |
| HTH | HHT | 2 to 1 |
| HTT | HHT | 2 to 1 |
| THH | TTH | 2 to 1 |
| THT | TTH | 2 to 1 |
| TTH | HTT | 3 to 1 |
| TTT | HTT | 7 to 1 |
If you want to play this in a bar then the way to remember the optimum is take the first player’s second choice, swap it (from H to T or vice versa) then add on his first two choices.