HMMT 二月 2014 · 冲刺赛 · 第 11 题

英文解析

11. [ 6 ] Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let N bethe remainder when the product of the numbers showing on the two dice is divided by 8. Find theexpected 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 numberand taking modulo 8 results in the same numbers, as all odd numbers are relatively prime to 8. Theexpected value in this case is 3 . 5.
    If the first die is even but not a multiple of 4, which has probability, then using similar reasoning,1
    N can be any of 0, 2, 4, 6 with equal probability, so the expected value is 3.4
    If the first die is 4, which has probability, then N can by any of 0, 4 with equal probability, so the 1
    expected value is 2.8
    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