HMMT 二月 2019 · COMB 赛 · 第 2 题

HMMT February 2019 — COMB Round — Problem 2

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled 1 , 2 , . . . , 20 on its sides). He conceals the results but tells you that at least half of the rolls are 20. Suspicious, you examine the first two dice and find that they show 20 and 19 in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show 20?

解析

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled 1 , 2 , . . . , 20 on its sides). He conceals the results but tells you that at least half of the rolls are 20. Suspicious, you examine the first two dice and find that they show 20 and 19 in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show 20? Proposed by: Evan Chen 1 Answer: 58 The given information is equivalent to the first two dice being 20 and 19 and there being at least two 20’s among the last three dice. Thus, we need to find the probability that given at least two of the last three dice are 20’s, all three are. Since there is only one way to get all three 20’s and 3 · 19 = 57 ways 1 1 to get exactly two 20’s, the probability is = . 1+57 58