HMMT 十一月 2012 · GEN 赛 · 第 4 题

HMMT November 2012 — GEN Round — Problem 4

[ 4 ] If you roll four fair 6-sided dice, what is the probability that at least three of them will show the same value?

解析

[ 4 ] If you roll four fair 6-sided dice, what is the probability that at least three of them will show the same value? 7 Answer: We have two cases: either three of the dice show one value and the last shows a 72 different value, or all four dice show the same value. In the first case, there are six choices for the value ( ) 4 of the dice which are the same and choice for which dice show that value. Then there are 5 choices 3 ( ) 4 for the last die. In total, there are 6 5 = 120 possibilities. For the second case, there are 6 values 3 120+6 126 7 that the last die can show. Consequently, the overall probability is, = = . 4 4 6 6 72