School Fair Spinner Game
At a school fair, a stall uses two fair spinners. Spinner X has four equal sectors labelled 1, 2, 3, 4. Spinner Y has three equal sectors labelled 1, 2, 3. A player pays ₹20 and spins both once. If the sum is 5 or more, the player wins ₹35; otherwise the player wins nothing. On the first day, the stall recorded 240 plays and paid out 96 prizes. The organiser wants to know whether the game is fair for the stall, meaning the expected profit per play is positive. A volunteer says the first-day result proves the game is unfair because only 96 of 240 won, while another says that is close to the theoretical probability. The organiser also wants the probability that a player gets a sum of exactly 4, and the probability of winning on two consecutive plays.
1. List the sample space for the sum or use a table to find P(sum ≥5) and P(sum=4).
Read the worked answer
There are 4×3=12 equally likely ordered outcomes. The sums are: (1,1)=2, (1,2)=3, (1,3)=4 (2,1)=3, (2,2)=4, (2,3)=5 (3,1)=4, (3,2)=5, (3,3)=6 (4,1)=5, (4,2)=6, (4,3)=7
Winning sums are 5 or more: (2,3), (3,2), (3,3), (4,1), (4,2), (4,3). That is 6 outcomes. P(sum≥5)=6/12=1/2.
Sum exactly 4 occurs for (1,3), (2,2), (3,1): 3 outcomes. P(sum=4)=3/12=1/4.
2. Calculate the expected payout and expected profit per play. Does the first-day result contradict the theoretical probability?
Read the worked answer
P(win)=1/2, and a win pays ₹35. Expected payout=0.5×35=₹17.50 per play. Player pays ₹20, so expected stall profit per play=20-17.50=₹2.50. The expected profit is positive, so the game is favourable to the stall in the long run.
First day: 96 wins in 240 plays gives 96/240=0.4. The theoretical probability is 0.5, so the observed proportion is lower than expected. This alone does not prove the game is unfair; random variation can produce fewer wins. It is a warning to keep records, not a mathematical proof.
3. Find the probability of winning on two consecutive plays and the stall’s expected profit over 240 plays. Compare this with the first-day actual profit.
Read the worked answer
Assuming plays are independent: P(win on two consecutive plays)=P(win)×P(win) =(1/2)×(1/2)=1/4.
Expected profit over 240 plays: 240×₹2.50=₹600.
First-day actual profit: Income=240×20=₹4,800. Prizes paid=96×35=₹3,360. Actual profit=4800-3360=₹1,440.
The actual first-day profit is higher than the expected ₹600 because there were fewer wins than expected. That is possible by chance; it does not by itself show the spinner is biased.