The trinomial distribution is a discrete probability distribution used to calculate probabilities in situations where a trial having three possible outcomes is repeated ‘n’ number of times.

It is a generalization of the binomial distribution which is used when a trial has exactly two outcomes.

**Example of a situation requiring the use of Trinomial Distribution**:

If a toss a coin 10 times then we use the binomial distribution to calculate the probability of getting a certain number of heads. This is because a coin toss has only two outcomes- heads or tails.

Now, suppose a game between 3 players A, B, and C is repeated 10 times. Here we use the trinomial distribution to calculate the associated probabilities since there are three possible outcomes for each game- player A wins, player B wins and player C wins.

**How to calculate probabilities using the Trinomial Distribution?**

We can calculate the probabilities by substituting the values of the random variables in the joint probability function of the distribution.

Let X denote the number of times the 1^{st} outcome occurs and let p_{1} denote the probability of the occurrence of the 1^{st} outcome.

Similarly, let Y denote the number of times the 2^{nd} outcome occurs and let p_{2} denote the probability of the occurrence of the 2^{nd} outcome.

If the total number of trials is ‘n’ then the third outcome clearly occurs “n-x-y” times.

The joint probability mass function of X and Y is given as,

**Example**:

Suppose a game between 3 players A, B, and C is repeated 12 times. Player A has a 50% chance of winning a game, player B has a 30% chance of winning the game, and Player C has a 20% winning probability.

Calculate the probability that player A wins 4 games, player B wins 3 games and player C wins the remaining 5 games.

**Solution**: Let X and Y denote the number of games won by players A and B respectively. We are given p_{1}=0.5, p_{2}=0.3, and p_{3}=0.2.

Substituting all this in the above formula we get,

P(X=4, Y=3) = (12! /4!*3!*5! )*(0.5)^{4}*(0.3)^{3}*(0.2)^{5 }= **0.0149688**.

**Marginal Probability distribution of X and Y**:

If we consider X as a random variable in its own right then we see that only two outcomes are possible- either 1^{st} outcome occurs or does not occur. Hence the marginal probability distribution of the random variable X is the binomial distribution with parameters n and p_{1}.

Similarly, the marginal probability distribution of the random variable Y is the binomial distribution with parameters n and p_{2}.

We can calculate the mean and variance of the variables X and Y jointly following the trinomial distribution by using the formula,

**Correlation between X and Y**:

It is clear that the two variables X and Y are not independent. Higher values of X imply lower values of Y since we know that the sum X+Y cannot exceed ‘n’. This means that X and Y are correlated (they are covariates).

In fact, we can calculate the correlation coefficient, which measures the linear relationship between X and Y. The correlation coefficient can be calculated using the formula,

**Example**:

Given (X, Y) jointly follows the trinomial distribution with parameters n=10, p_{1}=0.2, and p_{2}=0.5. Calculate the mean and variance of X and Y and the correlation coefficient between them.

**Solution**: We calculate the mean and variance using the above formula,

E(X) = np_{1}= 10*0.2 = 2

V(X) = np_{1}(1-p_{1})= 10*0.2*0.8 = 1.6

E(Y) = np_{2}= 10*0.5 = 5

E(X) = np_{2}(1-p_{2})= 10*0.5*0.5 = 2.5

The correlation coefficient is calculated as,

ρ = √(0.2*0.5/0.8*0.5) = √0.25 = 0.5

**Trinomial and Multinomial Distribution**:

The trinomial distribution is a particular case of the multinomial distribution. The multinomial distribution is used to calculate probabilities when a trial has three or more possible outcomes.

For example, when a dice is thrown there are 6 possible outcomes. We would use the multinomial distribution to calculate probabilities in this case.