HMMT 十一月 2012 · 冲刺赛 · 第 22 题
HMMT November 2012 — Guts Round — Problem 22
题目详情
英文原题
- [ 12 ] Alice generates an infinite decimal by rolling a fair 6-sided die with the numbers 1, 2, 3, 5, 7,
and 9 infinitely many times, and appending the resulting numbers after a decimal point. What is theexpected value of her number?
解析
英文解析
- [ 12 ]
∑
∞
1Xn
Answer: Let X be the n th number rolled. The number formed, 0 . X X · · · , is simply .
n 1 2 nn =1
2 10
( )
∑ ∑
∞ ∞ E ( X )
X nn
By linearity of expectation, the expected value is E = . However, the rolls aren nn =1 n =1
10 10
∑
∞
1 9 9 1 9 1 1
independent: for all n , E ( X ) = (1+2+3+5+7+9) = . So, our answer is · = · = .
n nn =1
6 2 2 10 2 9 2