PUMaC 2017 · 组合（B 组） · 第 5 题

PUMaC 2017 — Combinatorics (Division B) — Problem 5

There is a box containing 100 balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between 0 and 100, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can p be written in the form where p and q are relatively prime positive integers. Find p + q . q

解析

Let n = 100. By symmetry, the probability that the first ball to roll out is black is . The 2 k probability that there are exactly k black balls and the first ball to roll out is black is . n ( n +1) 2 k Thus, the probability that there are k black balls given that the first ball is black is . n ( n +1) 1 (Alternatively, note that it ought to be proportional to k .) Therefore, the probability there are k black balls and the second ball is black given that the first one is black is ( ) 2 k ( k − 1) k 4 = . n ( n + 1)( n − 1) 2 n ( n + 1)( n − 1) The final probability is ( ) ( ) n ∑ k 4 n + 1 4 2 = = . 2 n ( n + 1)( n − 1) 3 n ( n + 1)( n − 1) 3 k =2 Thus our answer is 2 + 3 = 5 . Problem written by Matt Tyler