Statistics - Multinomial Distribution

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

Where −

• ${n}$ = number of events

• ${n_1}$ = number of outcomes, event 1

• ${n_2}$ = number of outcomes, event 2

• ${n_x}$ = number of outcomes, event x

• ${P_1}$ = probability that event 1 happens

• ${P_2}$ = probability that event 2 happens

• ${P_x}$ = probability that event x happens

Example

Problem Statement:

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:

Given:

• ${n}$ = 12 (6 games total)

• ${n_1}$ = 1 (Player A wins)

• ${n_2}$ = 2 (Player B wins)

• ${n_3}$ = 3 (Player C wins)

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

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

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

Putting the values into the formula, we get: