## 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.