# 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

Pr=n!(n1!)(n2!)…(nx!)P1n1P2n2…PxnxPr=n!(n1!)(n2!)…(nx!)P1n1P2n2…Pxnx

Where −

·        nn = number of events

·        n1n1 = number of outcomes, event 1

·        n2n2 = number of outcomes, event 2

·        nxnx = number of outcomes, event x

·        P1P1 = probability that event 1 happens

·        P2P2 = probability that event 2 happens

·        PxPx = 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:

·        nn = 12 (6 games total)

·        n1n1 = 1 (Player A wins)

·        n2n2 = 2 (Player B wins)

·        n3n3 = 3 (Player C wins)

·        P1P1 = 0.20 (probability that Player A wins)

·        P1P1 = 0.30 (probability that Player B wins)

·        P1P1 = 0.50 (probability that Player C wins)

Putting the values into the formula, we get:

Pr=n!(n1!)(n2!)…(nx!)P1n1P2n2…Pxnx, Pr(A=1,B=2,C=3)=6!1!2!3!(0.21)(0.32)(0.53), =0.135