HMMT 二月 2016 · COMB 赛 · 第 2 题
HMMT February 2016 — COMB Round — Problem 2
题目详情
英文原题
- Starting with an empty string, we create a string by repeatedly appending one of the letters H , M ,
1 1 1
T with probabilities , , , respectively, until the letter M appears twice consecutively. What is the
4 2 4
expected value of the length of the resulting string?
解析
英文解析
- Starting with an empty string, we create a string by repeatedly appending one of the letters H , M ,
1 1 1
T with probabilities , , , respectively, until the letter M appears twice consecutively. What is the
4 2 4
expected value of the length of the resulting string?
Proposed by:
Answer: 6
Let E be the expected value of the resulting string. Starting from the empty string,
• We have a chance of not selecting the letter M ; from here the length of the resulting string is 1
1 + E .2
• We have a chance of selecting the letter M followed by a letter other than M , which gives a 1
string of length 2 + E .4
• We have a chance of selecting M twice, for a string of length 2.1
1 1 14
Thus, E = (1 + E ) + (2 + E ) + (2). Solving gives E = 6.
2 4 4