HMMT 二月 2005 · COMB 赛 · 第 8 题
HMMT February 2005 — COMB Round — Problem 8
题目详情
英文原题
- Every second, Andrea writes down a random digit uniformly chosen from the set
{ 1 , 2 , 3 , 4 } . She stops when the last two numbers she has written sum to a primenumber. What is the probability that the last number she writes down is 1?
解析
英文解析
- Every second, Andrea writes down a random digit uniformly chosen from the set
{ 1 , 2 , 3 , 4 } . She stops when the last two numbers she has written sum to a primenumber. What is the probability that the last number she writes down is 1?
Solution: 15 / 44
Let p be the probability that the last number she writes down is 1 when the firstnnumber she writes down is n . Suppose she starts by writing 2 or 4. Then she cancontinue writing either 2 or 4, but the first time she writes 1 or 3, she stops. Thereforep = p = . Suppose she starts by writing 1. Then she stops if she writes 1,1
2 4
2, or 4, but continues if she writes 3. Therefore p = (1 + p ). If she starts by 12
1 3
writing 3, then she stops if she writes 2 or 4 and otherwise continues. Therefore 4
1 1 1 3
p = ( p + p ) = (1 + 5 p ). Solving gives p = and p = . The probability we
3 1 3 3 3 1
4 16 11 11
1 15
want to find is therefore ( p + p + p + p ) = .
1 2 3 4
4 44