Bag of mice
time limit per test 2 seconds
memory limit per test 256 megabytes
Program Description
The dragon and the princess are arguing about what to do on the New Year’s Eve. The dragon suggests flying to the mountains to watch fairies dancing in the moonlight, while the princess thinks they should just go to bed early. They are desperate to come to an amicable agreement, so they decide to leave this up to chance.
They take turns drawing a mouse from a bag which initially contains w white and b black mice. The person who is the first to draw a white mouse wins. After each mouse drawn by the dragon the rest of mice in the bag panic, and one of them jumps out of the bag itself (the princess draws her mice carefully and doesn’t scare other mice). Princess draws first. What is the probability of the princess winning?
If there are no more mice in the bag and nobody has drawn a white mouse, the dragon wins. Mice which jump out of the bag themselves are not considered to be drawn (do not define the winner). Once a mouse has left the bag, it never returns to it. Every mouse is drawn from the bag with the same probability as every other one, and every mouse jumps out of the bag with the same probability as every other one.
Input
The only line of input data contains two integers w and b (0 ≤ w, b ≤ 1000).
Output
Output the probability of the princess winning. The answer is considered to be correct if its absolute or relative error does not exceed 10 - 9.
Sample test(s)
Input
1 3
Output
0.500000000
Input
5 5
Output
0.658730159
Note
Let’s go through the first sample. The probability of the princess drawing a white mouse on her first turn and winning right away is 1/4. The probability of the dragon drawing a black mouse and not winning on his first turn is 3/4 * 2/3 = 1/2. After this there are two mice left in the bag — one black and one white; one of them jumps out, and the other is drawn by the princess on her second turn. If the princess’ mouse is white, she wins (probability is 1/2 * 1/2 = 1/4), otherwise nobody gets the white mouse, so according to the rule the dragon wins.
解题心得：
- 题意就是一个箱子里面有w只白老鼠，b只黑老鼠。每回合公主从箱子中随机捉一只老鼠出来，然后龙从箱子中随机捉一只老鼠出来，然后箱子中的老鼠随机跑掉一只。谁先捉到白老鼠谁胜，如果都没捉到白老鼠并且箱子中已经没了老鼠龙胜。问公主胜利的概率是多少。
- 其实关于概率的部分很简单，上过高中的都知道。转移方程式也没有什么坑人的地方，直接上代码吧。
#include<bits/stdc++.h>
using namespace std;
const int maxn = 1010;
double dp[maxn][maxn];
int w,b;
int main()
{
while(cin>>w>>b)
{
memset(dp,0,sizeof(dp));
for(int i=1;i<=w;i++)
dp[i][0] = 1;
for(int i=1;i<=w;i++)
for(int j=1;j<=b;j++)
{
dp[i][j] += (double)i/(double)(i+j);
if(j >= 3)
dp[i][j] += (double)j/(double)(j+i)*(double)(j-1)/(double)(i+j-1)*(double)(j-2)/(double)(i+j-2)*dp[i][j-3];
if(j >= 2)
dp[i][j] += (double)j/(double)(j+i)*(double)(j-1)/(double)(i+j-1)*(double)(i)/(double)(i+j-2)*dp[i-1][j-2];
}
printf("%.9f
",dp[w][b]);
}
return 0;
}