HMMT 二月 2015 · COMB 赛 · 第 9 题
HMMT February 2015 — COMB Round — Problem 9
题目详情
- Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?
解析
- Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red? 9 Answer: Solution 1. The only information this gives us about the number of yellow balls 26 left is that it is even. A bijection shows that the probability that there are k yellow balls left is equal to the probability that there are 30 − k yellow balls left (flip the colors of the red and blue balls, and then switch the 65 balls that have been picked with the 65 balls that have not been picked). So the expected number of yellow balls left is 15. Therefore the expected number of red balls left is 22 . 5. So