HMMT 二月 2010 · 冲刺赛 · 第 16 题

英文解析

16. [ 9 ] Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset ofthem (such that each subset is equally likely) and puts them in a bag. You then draw three marblesfrom the bag with replacement. The colors you see are red, blue, red. What is the probability that theonly marbles in the bag are red and blue?
    Answer: There are two possible sets of marbles in the bag, { red,blue } and { red,blue,green } .27
    Initially, both these sets are equally likely to be in the bag. However, the probability of red, blue, red 35 − 3
    being drawn from a set S of marbles is proportional to | S | , as long as red and blue are both in S . By
    (1 / 2)3 − 3
    Bayes’s Rule, we must weight the probability of these two sets by | S | . The answer is .
    3 3
    (1 / 2) +(1 / 3)