HMMT 二月 2014 · 冲刺赛 · 第 11 题

HMMT February 2014 — Guts Round — Problem 11

[ 6 ] Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let N be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of N .

解析

[ 6 ] Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let N be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of N . 11 1 Answer: If the first die is odd, which has probability, then N can be any of 0, 1, 2, 3, 4 2 4, 5, 6, 7 with equal probability, because multiplying each element of { 0 , . . . , 7 } with an odd number and taking modulo 8 results in the same numbers, as all odd numbers are relatively prime to 8. The expected value in this case is 3 . 5. 1 If the first die is even but not a multiple of 4, which has probability, then using similar reasoning, 4 N can be any of 0, 2, 4, 6 with equal probability, so the expected value is 3. 1 If the first die is 4, which has probability, then N can by any of 0, 4 with equal probability, so the 8 expected value is 2. 1 1 Finally, if the first die is 8, which has probability, then N = 0. The total expected value is (3 . 5) + 8 2 1 1 1 11