Textbook Problem

Let X₁, X₂, X₃, X₄, X₅, and X₆ denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size n. Then these X_i’s have a multinomial distribution. According to the M&M Web site, the color proportions are p₁ = .24, p₂ = .13, p₃ = .16, p₄ = .20, p₅ = .13, and p₆ = .14.

1. a. If n = 12, what is the probability that there are exactly two M&Ms of each color?
2. b. For n = 20, what is the probability that there are at most five orange candies? [Hint: Think of an orange candy as a success and any other color as a failure.]
3. c. In a sample of 20 M&Ms, what is the probability that the number of candies that are blue, green, or orange is at least 10?