Betting a dollar on a seven

Question

A game charges \1 to play. You roll two fair dice; if they sum to exactly 7, you win \5 (gross), otherwise you win nothing. Your \1 is not returned. What is your expected net profit or loss per play?

Accepted Answer

First, the chance of rolling a 7. There are 6 ways out of 36: P(	ext{sum}=7) = \frac{6}{36} = \frac{1}{6}. Now the expected net result. You always pay \1; with probability 	frac16 you collect \5: E[	ext{net}] = -1 + \frac{1}{6}	imes 5 = -1 + \frac{5}{6} = -\frac{1}{6} \approx -0.167. You lose about 17 cents per play on average, so it's a bad bet. A fair payout would be \6 (since the true odds are 6-to-1); the \5 payout is the house's edge. Takeaway. Compare the payout to the "fair" payout implied by the odds: at \frac16 chance, break-even needs a \6 return on a \1 stake. Any payout below …

Betting a dollar on a seven

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