HMMT 十一月 2012 · GEN 赛 · 第 4 题

英文解析

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?
   Answer: We have two cases: either three of the dice show one value and the last shows a 7
   different value, or all four dice show the same value. In the first case, there are six choices for the value 72
   ( )
   of the dice which are the same and choice for which dice show that value. Then there are 5 choices 4
   ( )3
   for the last die. In total, there are 6 5 = 120 possibilities. For the second case, there are 6 values 4
   120+6 126 73
   that the last die can show. Consequently, the overall probability is, = = .
   4 4
   6 6 72