HMMT 十一月 2023 · 冲刺赛 · 第 8 题

HMMT November 2023 — Guts Round — Problem 8

[7] Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row.

解析

[7] Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row. Proposed by: Holden Mui Answer: 11/6 Solution: For each i = 1 , 2 , . . . , 6, let X denote the indicator variable of whether the i -th die shows i 1 the same number as the sixth die. Clearly, X = 1 always. For all other i , X is 1 with probability 6 i 6 1 and 0 otherwise, so E [ X ] = . By linearity of expectation, the answer is i 6 1 11 E [ X + · · · + X ] = E [ X ] + · · · + E [ X ] = 5 · + 1 = . 1 6 1 6 6 6