Charity Dice Game Expected Profit
At a school summer fair, a charity stall runs a dice game. A player pays £2 and rolls two fair six-sided dice. Under the current rules, a total of 7 wins £5, a total of 11 wins £3, and every other total wins nothing. The committee wants the stall to make an expected profit of at least £250 from 300 plays. It also has a rule that no individual prize may be more than £5, so the stall does not look like gambling with large payouts. A helper suggests adding a £10 prize for rolling a double six, believing this will attract more players. The stall leader must decide whether the current game already meets the profit target and whether the helper's change can be accepted under both the profit target and the prize cap.
1. Calculate the expected payout per play and the expected profit from 300 plays under the current rules. Does the current game meet the committee's target?
Read the worked answer
There are 36 equally likely outcomes for two dice. P(total 7) = 6/36 = 1/6. P(total 11) = 2/36 = 1/18. Expected payout = 5 × 1/6 + 3 × 1/18 = 5/6 + 1/6 = £1. Expected profit per play = £2 − £1 = £1. For 300 plays, expected profit = 300 × £1 = £300, which is above the £250 target.
2. Evaluate the helper's proposal exactly as stated. Include the new expected profit and check both the £250 target and the £5 prize cap.
Read the worked answer
P(double six) = 1/36. Adding a £10 prize changes expected payout to 1 + 10 × 1/36 = 1 + 0.2778 = £1.2778. Expected profit per play = 2 − 1.2778 = £0.7222, so for 300 plays = 300 × 0.7222 = £216.67, below £250. The £10 prize also breaks the £5 prize cap. Therefore the proposal cannot be accepted as stated.
3. The helper still wants a double-six prize. What is the largest whole-pound prize that keeps both the £5 prize cap and the £250 expected-profit target?
Read the worked answer
Let p be the double-six prize and keep the other prizes. Expected payout = 1 + p/36. Expected profit for 300 plays = 300(1 − p/36). Require 300(1 − p/36) ≥ 250 → 1 − p/36 ≥ 5/6 → p/36 ≤ 1/6 → p ≤ 6. The prize cap separately requires p ≤ 5. The largest whole-pound prize satisfying both is therefore £5. At £5, expected profit = 300(1 − 5/36) = 300 × 31/36 = £258.33, so the target is still met.