HMMT 二月 2026 · COMB 赛 · 第 2 题

HMMT February 2026 — COMB Round — Problem 2

Jacopo is rolling a fair 4 -sided die with faces labeled 1 , 2 , 3 , and 4 . He starts with a score of 0 . Every time he rolls a face with label i , he adds i to his score, and then replaces the label of that face with 0 . Compute Jacopo’s expected score after 4 rolls.

解析

Jacopo is rolling a fair 4 -sided die with faces labeled 1 , 2 , 3 , and 4 . He starts with a score of 0 . Every time he rolls a face with label i , he adds i to his score, and then replaces the label of that face with 0 . Compute Jacopo’s expected score after 4 rolls. Proposed by: Srinivas Arun 3 875 4 Answer: 10(1 − ( ) ) = 4 128 Solution: Let S be the total score contribution of the face initially labeled i . If this face is rolled i 3 4 at least once, then S = i . Otherwise, S = 0 . The probability that this face is never rolled is ( ) . i i 4 Therefore, ( ) ( ) 4 3 E [ S ] = i 1 − . i 4 By linearity of expectation, Jacopo’s expected score is E [ S + S + S + S ] = E [ S ] + E [ S ] + E [ S ] + E [ S ] 1 2 3 4 1 2 3 4 ( ) ( ) 4 3 = (1 + 2 + 3 + 4) 1 − 4 875 = 128