HMMT 十一月 2018 · 团队赛 · 第 1 题

HMMT November 2018 — Team Round — Problem 1

[ 15 ] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 .

解析

[ 15 ] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 . Proposed by: Michael Tang 31 Answer: 432 If any two of the dice show an odd number, then this is impossible, so at most one of the dice can show an odd number. We take two cases: Case 1: If exactly one of the dice shows an odd number, then all three other dice must show a multiple ( ) 3 1 1 1 of 4 , which can only be the number 4 . The probability that this occurs is 4 · · = . 2 6 108 Case 2: If all of the dice show even numbers, then the condition is satisfied. The probability that this ( ) 4 1 1 occurs is = . 2 16 1 1 31 The total probability is + = . 108 16 432