Polling organization probability

Question

A polling organization randomly selects an adult American for a survey about credit card usage. Suppose 47% of the population is male. Suppose 9.8% of males smoke cigars and 1.04% of females smoke cigars. If an adult is randomly selected for this survey who smokes cigars, what is the probability that person is a male?

Express your answer as a decimal, rounded to the nearest thousandth (three decimal places).

Answer 1

We are given 47% of the population is male. This means that 53% of the population must be female. We must now calculate the probability that a person is a man and smokes:

P(male ∩ smokes) = P(male) × P(smokes) = .47 × .098 = .04606

We must now calculate the probability that a person is female and smokes:

P(female ∩ smokes) = P(female) × P(smokes) = .53 × .0104 = .005512

We must now calculate the probability that a person smokes in general:

P(smokes) = P(female ∩ smokes) + P(male ∩ smokes) = .04606 + .005512 = 0.051572

The question asks P(male | smokes), thus:

P(male | smokes) = P(male ∩ smokes)/P(smokes)

P(male | smokes) = .04606/.051572

P(male | smokes) = .893