First to roll a six
Two players, A and B, take turns rolling a fair die, A first. The first player to roll a wins.
What is the probability that A wins?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Let be A's winning probability. If neither player rolls a in the first round, the game restarts in exactly the same position.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The game is the same after every full round with no six. Conditioning on the first round gives one linear equation.
Step 1: first round. A wins immediately with probability . Both players miss with probability , and then the game starts over. Otherwise B wins.
Step 2: equation. so
Another view. Look only at the first six to appear. If it appears at roll , A wins when is odd. The odds of A against B are , that is , so A wins with probability . Going first is worth about percentage points above a fair coin.