Charity Dice Game at the Science Fair
At the school science fair, the mathematics club runs a dice game to raise ₹300 for a library fund. In Game A, a player pays ₹10, rolls two fair six-sided dice, and wins ₹40 if the sum is 7 or 11. In Game B, a player pays ₹5, rolls one fair six-sided die, and wins ₹15 if the result is 5 or 6. The club expects about 400 plays if it chooses one game. The principal will allow the game only if the expected profit is at least ₹300 and the probability that a player wins is not more than 1/4. The treasurer prefers Game B because the prize is smaller, while the secretary prefers Game A because the entry fee is higher. The club cannot run both games at the same stall because of space and supervision.
1. Calculate the winning probability and expected profit per play for Game A.
Read the worked answer
For two dice, there are 36 equally likely outcomes.
Sum 7 occurs in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
Sum 11 occurs in 2 ways: (5,6), (6,5).
So the winning probability is
$$P(A)=\frac{8}{36}=\frac{2}{9}.$$
Expected payout $=40\times\frac{2}{9}=\frac{80}{9}\approx8.89$ rupees.
Expected profit per play $=10-8.89=1.11$ rupees.
For 400 plays, expected profit $\approx400\times1.11=444$ rupees, which is above ₹300.
2. Evaluate Game B against the principal's two conditions.
Read the worked answer
Game B wins on 5 or 6, so
$$P(B)=\frac{2}{6}=\frac{1}{3}.$$
This is greater than $\frac14$, so Game B violates the principal's probability condition even before profit is considered.
Its expected payout is $15\times\frac13=5$ rupees, equal to the ₹5 entry fee, so expected profit per play is ₹0. Over 400 plays the expected profit is ₹0, far below ₹300. Game B fails both conditions.
3. Which game should the club choose? Does the expected profit guarantee the ₹300 target?
Read the worked answer
Game A satisfies both conditions: $P(A)=\frac29\le\frac14$ and expected profit is about ₹444 over 400 plays. Game B fails both. The club should choose Game A.
However, expected profit is not a guarantee. If Game A is played 400 times and there are $W$ wins, profit is
$$4000-40W.$$
To reach ₹300, we need $4000-40W\ge300$, so $W\le92.5$, i.e. at most 92 wins. The expected number of wins is $400\times\frac29\approx88.9$, so the target is likely but not certain. The club should describe ₹444 as an expected value, not a promised amount.