sgu 495 Kids and Prizes

Title:

There are m people, n boxes, and each box initially contains a prize.

Each time, a person randomly draws one of the n boxes. If there is a prize in it, the prize is taken out; if not, it is put back intact.

Ask about the expectation of the number of prizes drawn.

Ideas:

At the beginning, when I think about people, I can't figure it out.

Each draw is independent. If you think about gifts, the probability of each gift being drawn is 1/n, and the probability of it not being drawn is (n-1)/n.

The probability that a gift is not drawn m times is ((n-1) / n) ^ m.

Then the expectation of the number left after n gifts are drawn m times is n * ((n-1) / n) ^ m, and the expectation of being drawn is also the expectation of the number obtained by the person is n - n * ((n-1)/n)^m.

Questions about probability should be thought more from the opposite side, thinking backwards, and thinking from different angles.

Code:

 1 #include <stdio.h>
 2 #include <string.h>
 3 #include <algorithm>
 4 using namespace std;
 5 double qp(double k,int n)
 6 {
 7     if (n == 0) return 1.0;
 8     double ans = qp(k,n / 2);
 9     ans *= ans;
10     if (n&1) ans *= k;
11     return ans;
12 }
13 int main()
14 {
15     int n,m;
16     while (scanf("%d%d",&n,&m) != EOF)
17     {
18         double ans;
19         ans = 1.0 * n * (1.0 - qp(1.0 * (n-1)/n,m));
20         printf("%.10f\n",ans);
21     }
22     return 0;
23 }