Statistics 简明教程

Statistics - Multinomial Distribution

多项式实验是一种统计实验，由 n 个重复试验组成。每个试验都有离散数量的可能结果。在任何给定的试验中，特定结果发生的概率都是恒定的。

A multinomial experiment is a statistical experiment and it consists of n repeated trials. Each trial has a discrete number of possible outcomes. On any given trial, the probability that a particular outcome will occur is constant.

Formula

其中——

Where −

  1. ${n}$ = number of events

  2. ${n_1}$ = number of outcomes, event 1

  3. ${n_2}$ = number of outcomes, event 2

  4. ${n_x}$ = number of outcomes, event x

  5. ${P_1}$ = probability that event 1 happens

  6. ${P_2}$ = probability that event 2 happens

  7. ${P_x}$ = probability that event x happens

Example

Problem Statement:

Problem Statement:

三个纸牌玩家进行一系列比赛。玩家 A 赢得任何一局游戏的概率为 20%，玩家 B 获胜的概率为 30%，而玩家 C 获胜的概率为 50%。如果他们玩 6 场比赛，玩家 A 赢得 1 场比赛、玩家 B 赢得 2 场比赛，而玩家 C 赢得 3 场比赛的概率是多少？

Three card players play a series of matches. The probability that player A will win any game is 20%, the probability that player B will win is 30%, and the probability player C will win is 50%. If they play 6 games, what is the probability that player A will win 1 game, player B will win 2 games, and player C will win 3?

Solution:

Solution:

给出：

Given:

  1. ${n}$ = 12 (6 games total)

  2. ${n_1}$ = 1 (Player A wins)

  3. ${n_2}$ = 2 (Player B wins)

  4. ${n_3}$ = 3 (Player C wins)

  5. ${P_1}$ = 0.20 (probability that Player A wins)

  6. ${P_1}$ = 0.30 (probability that Player B wins)

  7. ${P_1}$ = 0.50 (probability that Player C wins)

将这些值代入公式，即可得到：

Putting the values into the formula, we get: