riddle of the day

A game can last many rounds, but sometimes it is understood by looking only at the first cycle. If nothing happens at the start, the game returns exactly to where it began.

The First-Six Bet

Reasoner
Chance and uncertainty

Riddle statement

Two people take turns rolling a die.

The winner is the first person to roll a 6.

A rolls first. If A does not roll a 6, B rolls. If B also does not roll a 6, A rolls again, and so on.

What is the probability that A wins?

Show solution

Solution

Let p be the probability that A wins.

A can win immediately by rolling a 6 on the first roll. That happens with probability:

\frac{1}{6}

If A does not roll a 6 and B also does not roll a 6, then the game returns to the same initial state: it is A's turn again, with the same probability p of eventually winning.

The probability that both players fail in one round is:

\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36}

So:

p=\frac{1}{6}+\frac{25}{36}p

Subtract:

p-\frac{25}{36}p=\frac{1}{6}
\frac{11}{36}p=\frac{1}{6}
p=\frac{6}{11}

Answer: A wins with probability 6/11, approximately 54.5%.