If you and your opponent each roll a fair six-sided die, and you win $1 if your number is larger, what is your expected winning?

Free worked solution · Quant Blueprint

5/12 dollars if you receive $1 on a win and $0 otherwise.

1. There are 36 ordered outcomes. Six are ties.
2. Symmetry divides the remaining 30 outcomes equally between your wins and your opponent’s wins, so P(win) = 15/36 = 5/12.
3. The stated payoff is an indicator: its expectation is $1 × 5/12. No loss payment is specified.

Common mistake

Do not silently add a $1 loss when your opponent wins. That defines a different game with expected net payoff zero.

Follow-up

Conditioned on there being no tie, your win probability is (15/36)/(30/36) = 1/2. With an entry fee c, expected net profit is 5/12 − c dollars.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.