The Spinner Game Fairness Dispute
Two students design a spinner game for the school fair. The spinner has four equal sections labelled A, B, C, and D. The rules: spin once. If it lands on A, the player wins a prize. If it lands on B, the player wins nothing. If it lands on C or D, the player spins again. A parent complains the game is unfair because 'there are four outcomes, so the chance of winning must be 1 in 4.' The students run 200 trials and record 58 wins. They must decide whether the game matches the parent's claim and whether it is fair to players.
1. Use the rules to calculate the true probability of winning on a single play, and compare it with the parent's claim of 1 in 4.
Read the worked answer
On the first spin: A wins (probability 1/4), B loses (1/4), C or D means spin again (2/4 = 1/2).
The parent ignores the re-spin. A win can happen on the first spin or after any number of re-spins. Let P be the win probability. Then:
P = 1/4 + (1/2)P
Subtract (1/2)P: (1/2)P = 1/4, so P = 1/2.
Check by listing: win on spin 1 with 1/4; win on spin 2 with (1/2)(1/4) = 1/8; spin 3 with 1/16, and so on. The sum 1/4 + 1/8 + 1/16 + … = 1/2.
The true win probability is 1/2, not 1/4. The parent counted outcomes without accounting for the re-spin rule.
2. The 200 trials produced 58 wins. Decide whether this result is consistent with your calculated probability, and comment on whether the game is fair to players.
Read the worked answer
If the true probability is 1/2, then in 200 trials we expect about 100 wins. Getting 58 is far below 100, a gap of 42 wins.
A rough check: with 200 trials, results typically vary by only about 7 wins from the expected 100 (the spread is roughly the square root of 200 × 1/2 × 1/2, about 7). A result of 58 is many spreads away from 100, so it is very unlikely to come from a fair 1/2 spinner. Either the spinner is not balanced, or the trials were recorded incorrectly, or the re-spin rule was not followed.
On fairness: a 1/2 win chance means half of all plays win a prize, which is generous to players if prizes cost money. The data suggest the actual game is much stingier than designed, so the students should inspect the spinner and re-run trials before the fair. Concluding 'the probability is wrong' is not justified; the design probability stands, but the physical spinner may be faulty.