A number of distribution is defined
A randomized trial if there \ (k \) possible outcomes \ (A_1, A_2, \ cdots, a_k \) , respectively, their number of occurrences recorded as random variables \ (X_1, X_2, \ cdots, X_k \) , they the probability distribution are \ (p_1, P_2, \ cdots, p_k \) , then \ (n \) the overall results of samples in, \ (A_1 \) appears \ (n_1 \) times, \ (A_2 \) appears \ (n_2 \) times, ..., \ (a_k \) appears \ (n_k \) appears the probability of such an event once \ (P \) has the following formula:
| \ (k \) possible outcomes | \(A_1\) | \(A_2\) | \(\cdots\) | \(A_k\) | |
|---|---|---|---|---|---|
| The number of times each result appears | \(X_1\) | \(X_2\) | \(\cdots\) | \(X_k\) | |
| The probability of each possible result | \(p_1\) | \(p_2\) | \(\cdots\) | \(p_k\) | |
| Sampling \ (n \) times | |||||
| \(n_1\) | \(n_2\) | \(\cdots\) | \(n_k\) | \(\sum_{i=1}^n n_i = n\) | |
| \(x_1\) | \(x_2\) | \(\cdots\) | \(x_k\) | \(\sum_{i=1}^n x_i = n\) |
\[\bm{P}(X_1=n_1,X_2=n_2,\cdots,X_k=n_k)=\frac{n!}{n_1!n_2!\cdots n_k!}p_1^{n_1}p_2^{n_2}\cdots p_k^{n_k}\]