Question

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 Xi's have a multinomial distribution. Suppose it is claimed that the color proportions are p1 = 0.21, p2 = 0.13, p3 = 0.19, p4 = 0.2, p5 = 0.12, and p6 = 0.15.

(a) If n = 12, what is the probability that there are exactly two M&Ms of each color? (Round your answer to four decimal places.)

(b) For n = 20, what is the probability that there at most five orange candies? [Hint: Think of an orange candy as a success and any other color as a failure.] (Round your answer to three decimal places.)

(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 9? (Round your answer to three decimal places.)

Answer 1

a)P(2 from each )=(12!/(2!*2!*2!*2!*2!*2!))*(0.21*0.13*0.19+0.2*0.12*0.15)^2

=0.0026

b)

P(X<=5)=

∑x=0a     (nCx)px(1−p)(n-x)    =

0.804

c)

P(X>=9)=1-P(X<=8)=

1-∑x=0x-1   (nCx)px(q)(n-x) =

0.943