D. Bag of mice
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
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.
Examples
input
Copy
1 3output
Copy
0.500000000input
Copy
5 5output
Copy
0.658730159Note
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.
解析：
公主在i只白鼠，j只黑鼠情况下能过获胜的所有情况包括：
1、 1.0*i/(i+j); 公主当前先拿到白鼠
2、 1.0*j/(i+j) * (j-1)/(i+j-1); 公主没拿到白鼠，但下一次龙也没拿到
3、 1.0*i/(i+j-2); 龙没拿到的同时放走一只白鼠
4、 dp[i-1][j-2] 在失去两只黑鼠一只白鼠情况下获胜
5、 1.0*(j-2)/(i+j-2); 龙没拿到的同时放走一只黑鼠
6、 dp[i][j-3] 在失去三只黑鼠情况下获胜
#include<bits/stdc++.h>
#define IO ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define rep(i,j,k) for(int i=j;i<k;i++)
const int maxn = 1e3+5;
double dp[maxn][maxn];
using namespace std;
int main(void)
{
IO
int w,b;
cin>>w>>b;
//rep(i,0,b+1) dp[0][i] = 0;
rep(i,1,w+1) dp[i][0] = 1,dp[i][1] = 1.0*i/(i+1),dp[i][2] = 1.0*i/(i+2) + 1.0*2/(i+2)*1/(i+1);
rep(i,1,w+1)
rep(j,3,b+1)
dp[i][j]=1.0*i/(i+j) + 1.0*j/(i+j)*(j-1)/(i+j-1)*( dp[i-1][j-2] * 1.0*i/(i+j-2) + dp[i][j-3] * 1.0*(j-2)/(i+j-2)) ;
/*
公主在i只白鼠，j只黑鼠情况下能过获胜的所有情况包括：
1.0*i/(i+j); 公主当前先拿到白鼠概率
1.0*j/(i+j) * (j-1)/(i+j-1); 公主没拿到白鼠，但下一次龙也没拿到
1.0*i/(i+j-2); 龙没拿到的同时放走一只白鼠
dp[i-1][j-2] 在失去两只黑鼠一只白鼠情况下获胜
1.0*(j-2)/(i+j-2); 龙没拿到的同时放走一只黑鼠
dp[i][j-3] 在失去三只黑鼠情况下获胜
*/
cout<<fixed<<setprecision(10)<<dp[w][b]<<endl;
}