Question

a. Consider 5 fish in a bowl: 3 of them are red, and 1 is green, and 1 is blue. Select the fish one at a time, without replacement, until the bowl is empty. Let X=1 if all of the red fish are selected, before the green fish is selected; and X=0 otherwise. Find E(X). (Hint: You already found pX(1) on Monday, and pX(0) is just the complementary probability.)

b. Suppose that 60% of people in Chicago are fans of da Bears. Assume that the fans' preferences are independent. We interview 3 people from Chicago, and we let X denote the number of fans of da Bears. Find E(X).

pX(0)=

pX(1)=

pX(2)=

pX(3)=

pX(x)=0 otherwise.

E(X)=

Answer 1

a) X = 1, if all the red fish are slected before the green is selected.
The probability of this could be computed as:

P(X = 1) = Probability that the first 3 fish drawn are red + Probability that the first 3 fish drawn are such that there are 2 red and 1 blue fish * Probability that the 4th fish drawn is red

Therefore the expected value of X here is computed as:

Therefore 0.25 is the required expected value here.

b) The number of fans of Da beard could be modelled here as:

The required probabilities are computed here using the binomial probability function as:

Therefore the expected value is now computed here as:

Therefore 1.8 is the required expected value here.