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 probabipty 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

${P_1}$ = probabipty that event 1 happens

${P_2}$ = probabipty that event 2 happens

${P_x}$ = probabipty that event x happens

Example

Problem Statement:

Three card players play a series of matches. The probabipty that player A will win any game is 20%, the probabipty that player B will win is 30%, and the probabipty player C will win is 50%. If they play 6 games, what is the probabipty 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 (probabipty that Player A wins)

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

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

Putting the values into the formula, we get: