Question

It is estimated that, of the people viewing a certain movie over the weekend, 45 percent were adult women, 42 percent were adult men, and 13 percent were children. If we randomly interview people at a theatre about the movie, each interview takes a few minutes, so we will not interview people in the same group or family. So we may assume that their gender and age classification is independent from person to person. 1a. What is the probability that the first five people we interview are all adults? 1b. What is the probability that the first adult we interview is a female?

Answer 1

(1a) The probability that the first five people we interview are all adults is 0.4984.

(1b) The probability that the first adult we interview is a female is 0.5172.

Explanation:

Denote the events as follows:

F = a female viewer

M = a male viewer

A = an adult viewer

C = children

The information provided is:

P (A ∩ F) = 0.45

P (A ∩ M) = 0.42

P (C) = 0.13

According to the law of total probability the probability of an event X is given as follows:

P (X) = P (X ∩ Y) + P (X ∩ Y')

Use this formula to compute the probability of event A as follows:

P (A) = P (A ∩ F) + P (A ∩ M)

= 0.45 + 0.42

= 0.87

(1a)

Compute the probability that the first five people we interview are all adults as follows:

P (First 5 adults) = [P (A)]⁵

= [0.87]⁵

= 0.4984209

≈ 0.4984

Thus, the probability that the first five people we interview are all adults is 0.4984.

(1b)

The conditional probability of an event X given that another event Y has already occurred is:

P (X | Y) = P (X ∩ Y) ÷ P (Y)

Compute the probability that the first adult we interview is a female as follows:

P (F | A) = P (A ∩ F) ÷ P (A)

= 0.45 ÷ 0.87

= 0.51724137

≈ 0.5172

Thus, the probability that the first adult we interview is a female is 0.5172.