Question

A random sample of adult drivers was obtained, 53% men and 46% women. A survey showed that 57% of the drivers rely on GPS systems. 32% of the drivers are men and use GPS while 24% of the drivers are women and use GPS. Suppose a person included in this survey is randomly selected.

a.) Suppose the person selected is a man. What is the probability that he relies on a GPS system? Your answer should have at least 3 decimal places.

b.) Suppose the person selected relies on a GPS system. What is the probability that the person is a woman? Your answer should have at least 3 decimal places.

c.) What is the probability that the person is a man and does not rely on a GPS system? Your answer should have at least 3 decimal places.

d.) What is the probability that an individual is a man or uses a GPS system? Your answer should have at least 3 decimal places.

e.) What is the probability that an individual does not use a GPS system? Your answer should have at least 3 decimal places.

Answer 1

Let M shows the event that person is man and W shows the event that person is woman. So we have

P(M) = 0.54, P(W) = 0.46

Let G shows the event that person uses GPS. So we have

P(G) = 0.57, P(G') = 1 - P(G) = 1 - 0.57 = 0.43

And we have

P(G and M) = 0.32, P(G and W) = 0.24

(a)

The probability that person relies on a GPS system, given that person has man is

P(G | M) = P(G and M) / P(M) = 0.32 / 0.54 = 0.5926

Answer: 0.5926

(b)

The probability that the person is a woman, given that person relies on GPS is

P(W | G) = P(G and W) / P(G) = 0.24 / 0.57 = 0.4211

Answer: 0.4211

(c)

P(M and G') = P(M) - P(M and G) = 0.54 - 0.32 = 0.22

Answer: 0.220

(d)

P(M or G) = P(M) + P(G) - P(M and G) = 0.54 + 0.57 - 0.32 = 0.79

Answer: 0.790

(e)

By the complement rule,

P(G') = 1 - P(G) = 1 - 0.57 = 0.43

Answer: 0.430