Question

Pollsters are concerned about declining levels of cooperation among persons contacted in surveys. A pollster contacts 8686 people in the​ 18-21 age bracket and finds that 4343 of them respond and 4343 refuse to respond. When 276276 people in the​ 22-29 age bracket are​ contacted, 258258 respond and 1818 refuse to respond. Suppose that one of the 362362 people is randomly selected. Find the probability of getting someone in the 18 dash 2118-21 age bracket or someone who respondedresponded.

Answer 1

0.9503

Explanation:

First of all, there are some wrong figures in the original text. Because there is a total of 362362 people, the figures should be 86086 (people in the 18-21 age bracket), 43043 (people in the 18-21 age bracket who respond) and 43043 people in the 18-21 age bracket who refuse to respond. In the same way, because there are 276276 people in the 22-29 age bracket, it should be 18018 and not 1818 who refuse to respond in this subset of people. Now, let's define the following events:

R: a person respond

A: a person belongs to the 18-21 age bracket. So,

The number of people who respond is 43043 + 258258 = 301301, so

P(R) = 301301/362362 = 0.8315

P(A) = 86086/362362 = 0.2376

P(R | A) = 43043/86086 = 0.5

We are looking for P(A∪R) = P(A) + P(R) - P(A∩R),

P(A∩R) = P(R | A)P(A) = (0.5)(0.2376) = 0.1188, so,

P(A∪B) = 0.2376 + 0.8315 - 0.1188 = 0.9503