Problem of the Day #12

POTD December 1, 2023
Author: munch

Munch rigs a coin to land heads with probability p. Then, Munch flips the coin until two consecutive heads are flipped. If the total number of flips is odd, Munch wins p dollars; otherwise, Munch wins nothing. Let q be the value of p that Munch should pick to maximize his expected win. If q = a-\sqrt{b}, where a, b \in \mathbb{Z} and \sqrt{b} cannot be simplified, find a^b.