Analysis of The Three-bit Game
For the three-bit sequence game, the second player can optimise his odds by choosing sequences according to:
| 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 |
In each case the second player's optimal choice is to precede the first player's sequence (underlined) with the opposite of the second symbol (bolded).
An easy way to remember the sequence for using as a bar trick is for the second player to use the first two choices of the first player as the last choices, and then using the opposite of the (now) last choice as the first one.
- So for the first player's choice of XYZ
- the second player must choose xXY
where x - is the opposite of Y, and XY are the same XY chosen by the first player.
An intuitive explanation for this result, is that in any case that the sequence is not immediately the first player's choice, the chances for the first player getting their sequence-beginning, the opening two choices, are usually the chance that the second player will be getting their full sequence. So the second player will most likely "finish before" the first player.
Read more about this topic: Penney's Game
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