## Exercises - Multinomial Distribution

1. Suppose a card is drawn randomly from an ordinary deck of playing cards, and then put back in the deck. This is repeated five times. What is the probability of drawing 1 spade, 1 heart, 1 diamond, and 2 clubs?

$$\frac{5!}{1! 1! 1! 2!} (.25)^1 (.25)^1 (.25)^1 (.25)^2 \doteq 0.05859$$

2. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue marbles. We randomly select 4 marbles from the bowl, with replacement. What is the probability of selecting 2 green marbles and 2 blue marbles?

$$\frac{4!}{0! 2! 2!} (0.2)^0 (0.3)^2 (0.5)^2 = 0.135$$

3. In a certain town, 40% of the eligible voters prefer candidate A, 10% prefer candidate B, and the remaining 50% have no preference. You randomly sample 10 eligible voters. What is the probability that 4 will prefer candidate A, 1 will prefer candidate B, and the remaining 5 will have no preference?

$$\frac{10!}{4!1!5!} (.40)^4 (0.10)^1 (0.50)^5 \doteq 0.1008$$