hdu 4405 Aeroplane chess

Problem Description

Hzz loves aeroplane chess very much. The chess map contains N+1 grids labeled from 0 to N. Hzz starts at grid 0. For each step he throws a dice(a dice have six faces with equal probability to face up and the numbers on the faces are 1,2,3,4,5,6). When Hzz is at grid i and the dice number is x, he will moves to grid i+x. Hzz finishes the game when i+x is equal to or greater than N.

There are also M flight lines on the chess map. The i-th flight line can help Hzz fly from grid Xi to Yi (0
Please help Hzz calculate the expected dice throwing times to finish the game.

 

Input

There are multiple test cases.
Each test case contains several lines.
The first line contains two integers N(1≤N≤100000) and M(0≤M≤1000).
Then M lines follow, each line contains two integers Xi,Yi(1≤Xi The input end with N=0, M=0.

 

Output

For each test case in the input, you should output a line indicating the expected dice throwing times. Output should be rounded to 4 digits after decimal point.

 

Sample Input

2 0 8 3 2 4 4 5 7 8 0 0

 

Sample Output

1.1667 2.3441

题目大意：求飞行棋从0到n的期望掷骰子点数，其中还有xi可以直接飞到yi的点，可以连飞。

思路：虽然不是很难，第一道自己写的概率dp还是应该纪念一下。求期望用逆推的方式，dp[i]表示从点i到n还需要的期望。

那么独立的考虑当前点i，要么可以直接飞走（期望直接等于目的地的期望），要么等概率1/6地去前6个点。转移方程为：

dp[i]=1/6(dp[i+1]+1) + 1/6(dp[i+2]+1) + ..... + 1/6( dp[i+6]+1)

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