Two Lotteries: Expected Value vs. Chance of Winning
At a festival, a student can buy at most one ticket of each type with her 1,000 yen. Lottery A: a 300-yen ticket gives one draw with probability 0.05 of winning 3,000 yen; otherwise nothing. Lottery B: an 800-yen ticket gives three independent draws, each with probability 0.04 of winning 3,000 yen, and multiple wins are paid separately. She says she wants 'the best chance of not going home empty-handed,' while her friend insists that 'expected value is the only rational criterion.' She cannot buy fractional tickets, cannot resell prizes, and cannot combine the two tickets because 300 + 800 exceeds her budget. She must decide before the stall closes in ten minutes.
1. Compute the expected net gain and the probability of at least one prize for each lottery. Which is better under each criterion?
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Lottery A: expected net gain = 0.05 × 3,000 − 300 = 150 − 300 = −150 yen. Probability of at least one prize = 0.05.
Lottery B: expected net gain = 3 × 0.04 × 3,000 − 800 = 360 − 800 = −440 yen. Probability of at least one prize = 1 − 0.96³ = 1 − 0.884736 ≈ 0.1153, about 11.5%.
On expected value, A is less bad (−150 versus −440). On 'not going home empty-handed', B is more than twice as likely to yield at least one prize (11.5% versus 5%). The two criteria point in opposite directions, and neither ticket has positive expected value.
2. Which should she choose? Justify using decision criteria, not just the larger number.
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The choice depends on her objective, so the honest answer is conditional. If she is risk-neutral and cares about average money, A is better: it loses 150 yen on average versus 440 yen for B. If her stated goal is genuinely the probability of winning anything, B is better, at the cost of a much worse average loss and a higher ticket price.
The friend's claim that expected value is 'the only rational criterion' is too strong. Expected value suits repeated decisions with small stakes; for a one-off purchase where the experience of winning has utility beyond the prize, maximizing win probability can be rational. A defensible recommendation: choose A if the 300-yen cost matters and she wants the better average; choose B only if she values the 11.5% chance enough to pay for it.
3. She considers buying both tickets. Check feasibility, then compute the hypothetical combined expected net gain and probability of at least one prize.
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Buying both is infeasible: 300 + 800 = 1,100 yen, which exceeds her 1,000-yen budget, and combining is excluded.
Hypothetically, the combined expected net gain would be −150 + (−440) = −590 yen, and the probability of at least one prize would be 1 − (0.95)(0.96³) ≈ 1 − 0.8405 = 0.1595, about 16.0%.
Even if it were affordable, it would be worse in expected value than either ticket alone, so the budget constraint is not the only reason to reject it; the arithmetic also argues against it. The exam-style judgement is to check feasibility first, then compute only for feasible options.