PUMaC 2015 · 组合（A 组） · 第 2 题

PUMaC 2015 — Combinatorics (Division A) — Problem 2

[ 3 ] Andrew has 10 balls in a bag, each a different color. He randomly picks a ball from the bag 4 times, with replacement. The expected number of distinct colors among the balls he picks p is , where gcd( p, q ) = 1 and p, q > 0. What is p + q ? q

解析

[ 3 ] Andrew has 10 balls in a bag, each a different color. He randomly picks a ball from the bag 4 times, with replacement. The expected number of distinct colors among the balls he picks p is , where gcd( p, q ) = 1 and p, q > 0. What is p + q ? q ( ) 4 9 Solution: The probability that any particular one of the 10 colors is picked is p = 1 − = 10 3439 . The expected contribution towards the total number of distinct colors picked by any 10000 particular color is then p · 1 + (1 − p ) · 0 = p , and by linearity of expectation, since we have 3439 10 colors, the expected total number of distinct colors is E = 10 · p = , so p = 3439 and 1000 q = 1000 and p + q = 4439 . Author: Roy Zhao