Q6) A company produces candies in three colors: red, blue, and green. From past data, it’s known that 40% of candies are red, 30% are blue, and 30% are green. If you randomly select 10 candies from a large bin, what is the probability that you get 4 red candies, 3 blue candies, and 3 green candies?
☞ The multinomial distribution is a generalization of the binomial, i.e instead of only telling you the probability of seeing successes (where the only other outcome is failure) it tells you the probabilities of observing counts among multiple categories. ☞ Therefore, the binomial distribution is just a special case of the multinomial distribution, where the only 2 categories are success/fail.
Parameters:
- : the number of trials
- : the probabilities of each outcomes (i.e for a dice roll, each should be )
Probability Mass Function (PMF): The probability of observing a particular set of outcomes is given by:
Where:
- ( X_i ) is the number of trials that result in outcome ( i )
- ( x_i ) is a specific number of successes for outcome ( i ) in ( n ) trials
Returning to our candy example:
Here, ( n = 10 ) (as you pick 10 candies), ( p_{red} = 0.4 ), ( p_{blue} = 0.3 ), and ( p_{green} = 0.3 ). You want ( x_{red} = 4 ), ( x_{blue} = 3 ), and ( x_{green} = 3 ).
Plugging the values into the PMF:
[ P(X_{red} = 4, X_{blue} = 3, X_{green} = 3) ] [ = \frac{10!}{4!3!3!} \times 0.4^4 \times 0.3^3 \times 0.3^3 ]
Evaluating this gives you the desired probability.
By understanding and using the multinomial distribution, you can determine the likelihood of specific combinations of outcomes for multi-category experiments, and should really give you a better grasp of the binomial distribution.