Solving the Penney Ante Problem with Markov Chains, Martingales, and a Combinatoric Approach
Imagine a two player game where each player is assigned a sequence, for example THTH and HTHH, and a coin is flipped until either player sees their sequence. The first player to see their sequence appear wins. Given two sequences, which sequence is expected to come first in the sequence of coin flips? What is the probability of a certain player winning? Although these two questions sound similar, the result is that in our example, the expected number of turns for sequence A to appear is 20 and for B it is 18. Meanwhile, the probability of A winning is 9/14 while the probability of B winning is 5/14. Additionally the game has the property that for any sequence A chooses, B can always find a sequence that has a higher probability of winning. These counterintuitive results are the core of the Penney Ante problem, discovered by Walter Penney [3]. My thesis will study this problem through three approaches based in Markov chains, martingales, and a combinatoric approach.
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