HMMT 十一月 2014 · 冲刺赛 · 第 30 题
HMMT November 2014 — Guts Round — Problem 30
题目详情
- [ 15 ] Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1 , 2 , . . . , 2014) until weobtain a value less than or equal to the previous roll. Let E be the expected number of times we rollthe die. Find the nearest integer to 100 E .
HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND
Organization Team Team ID#
英文原题
[ 15 ] Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1 , 2 , . . . , 2014) until we
obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll
the die. Find the nearest integer to 100 E .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 15 ] Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1 , 2 , . . . , 2014) until weobtain a value less than or equal to the previous roll. Let E be the expected number of times we rollthe die. Find the nearest integer to 100 E .
Answer: 272 Let n = 2014. Let p denote the probability the sequence has length at least k . Wekobserve that
( )
p =knksince every sequence of k rolls can be sorted in exactly one way. Now the answer isnk
( )
∑np = 1 + .1
k ≥ 0 nk
As n → ∞ , this approaches e . Indeed, one can check from here that the answer is 272.
Guts Round