Monte Carlo Simulations: Penney’s Game

An n-gram is a word of length n that we can create from a set of letters. With two letters \{\mbox{T}, \mbox{H}\}, we can form these 3-grams: \mbox{TTT}, \mbox{TTH}, \mbox{THT}, \mbox{THH}, \mbox{HTT}, \mbox{HTH}, \mbox{HHT}, and \mbox{HHH}. We will let \mbox{T} stand for tails and \mbox{H} for heads, so that a 3-gram can represent an outcome if we flip a coin 3 times.

Penney’s game concerns two good friends and their n-grams. The game is simple and seemingly fair. Soon Player 1 will call off their long friendship and leave, however. We will find that, whenever n \geq 3, Player 2 is always more likely to win.

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