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Probability, Statistics and Queueing

There are 1,000 lottery tickets at ¥50 each: one wins a ¥10,000 prize, ten win ¥1,000 prizes, and the rest win nothing. Which of the following is the expected value of the prize per ticket?

  1. A−¥30
  2. B¥11
  3. C¥20
Show answer and explanation

Answer: C. ¥20

The expected value is the sum of each value multiplied by the probability of that value. The probability of ¥10,000 is 1/1,000 and that of ¥1,000 is 10/1,000, so 10,000 × 1/1,000 + 1,000 × 10/1,000 = 10 + 10 = ¥20. Subtracting the ticket price of ¥50 gives an expected gain or loss of −¥30 per ticket.

  • A. This is the expected gain or loss, the expected prize of ¥20 minus the ticket price of ¥50. The question asks for the expected value of the prize.
  • B. This is (10,000 + 1,000) ÷ 1,000 = 11, which ignores that there are ten ¥1,000 prizes.
  • C. Correct. (10,000 × 1 + 1,000 × 10) ÷ 1,000 = ¥20.
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