Game

Alice and Bob play the following game. First, they collect $N$ small stones and put them in one pile. Then they take turns, beginning with Alice, and throw a coin. If a player throws heads, they take a heap of stones; if tails, then they do not take it. The winner is the one who takes the last stone. The players are known to have the ability to throw the desired side of a coin. More precisely, if Alice wants to throw her preferred side of the coin, then she achieves it with probability $P$. The same probability for Bob is $Q$. It is required to find the probability with which Alice will win the game, provided that both players play optimally.