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Penney's Ante is a coin tossing game for two people.
Player 1 chooses a sequence of three results, say THT,
and Player 2 chooses another sequence, say TTH,
and the coin is flipped until one of the sequences appears, for example
THHTHTHTHHTHHTTTH
In the above sequence the game would end after
THHTHT
with Player 1 winning as his sequence appears first.
There are 8 possible sequences to choose from, HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Now it appears the game is completely random but in fact Player 2 has a higher chance of winning if he follows the rule to always start sequence with the reverse of Player 1's middle choice, followed by the first two choices of Player 1, which is what I did in this example.
I came across this reading Matt Parker's book Humble Pi.
The chance of Rod winning is shown in the table, and the chance of Matt winning is (1-Rod's), and it can be seen that Matt's chance is at worse twice that of Rod's. In the book Matt sugests his readers do not worry about this column, but I did and wanted to know how this was calculated. I always find problems with probability amongst the most difficult. This was originally documented by Walter Penney in 1969, hence the name.
To this end I discovered a YouTube video from which I was able to understand the method. The notes below are my workings following the approach in this video..
Note that in a simple coin tossing whatever went before does not effect the outcome of the next toss. So if previous flips were HHHH then the possibility of the next toss being H is still 1/2. Many people believe this not to be so, and confuse it with the possibility of getting 5 heads in a row before any is taken is (1/2)^5 or 1/32.
In the Penney's Ante game the last two results of the previous flips become the first two of the next sequence of three. The effect of this can be seen in the following notes.
Penney’s Ante.pdf
Screen capture from YouTube of all the possibilities for Player 2 winning when Player 1 chooses HTH
Completed chart with all the possibilities of Player 2 winning.
Best choice for Player 2 winning for each possibilities available to Player 1. Note as stated at the outset this is when Player 2 chooses the reverse of the middle of Player 1 followed by the first two of Player 1.
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