HMMT 十一月 2012 · 冲刺赛 · 第 22 题
HMMT November 2012 — Guts Round — Problem 22
题目详情
- [ 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 the expected value of her number?
解析
- [ 12 ] ∑ ∞ X 1 n Answer: Let X be the n th number rolled. The number formed, 0 . X X · · · , is simply . n 1 2 n n =1 2 10 ( ) ∑ ∑ ∞ ∞ E ( X ) X n n By linearity of expectation, the expected value is E = . However, the rolls are n n n =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 n n =1 6 2 2 10 2 9 2