HMMT 十一月 2013 · GEN 赛 · 第 4 题

HMMT November 2013 — GEN Round — Problem 4

[ 4 ] A 50-card deck consists of 4 cards labeled “ i ” for i = 1 , 2 , . . . , 12 and 2 cards labeled “13”. If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label?

解析

[ 4 ] A 50-card deck consists of 4 cards labeled “ i ” for i = 1 , 2 , . . . , 12 and 2 cards labeled “13”. If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label? 73 Answer: All pairs of distinct cards (where we distinguish cards even with the same label) are 1225 ( ) ( ) ( ) 2 4 50 49 equally likely. There are + 12 = 73 pairs of cards with the same label and = 100 · = 1225 2 2 2 4 73 pairs of cards overall, so the desired probability is . 1225