Question

On a recent holiday evening, a sample of 500 drivers was stopped by the police. Three hundred were under 30 years of age. A total of 250 were under the influence of alcohol. Of the drivers under 30 years of age, 200 were under the influence of alcohol.

Let A be the event that a driver is under the influence of alcohol.
Let Y be the event that a driver is less than 30 years old.
Show the joint probability table.
Are A and Y mutually exclusive events? Explain.
Are A and Y independent events? Explain.

Answer 1

A and Y are neither mutually exclusive nor independent events. They are not mutually exclusive because some individuals are both under 30 and under the influence of alcohol. They are not independent because the probability of their joint occurrence is not the product of their individual probabilities.

Step-by-step explanation:

The joint probability table compares the event A (driver is under the influence of alcohol) with the event Y (driver is less than 30 years old) based on the provided information. To answer whether A and Y are mutually exclusive or independent events, let's examine the definition and data:

Mutually exclusive events cannot happen at the same time. In this scenario, since there are drivers under 30 who are also under the influence of alcohol, A and Y are not mutually exclusive because both events do happen simultaneously for some individuals.

Independent events are such that the occurrence of one event does not affect the probability of the other. To determine this, we need to check whether P(A and Y) = P(A)P(Y). Based on the numbers provided, P(A and Y) = 200/500 = 0.40, whereas P(A) = 250/500 = 0.50 and P(Y) = 300/500 = 0.60. Therefore, P(A and Y) is not equal to P(A)P(Y), indicating that A and Y are not independent.

Joint Probability Table
Y
Not Y
Total


A
200
50
250


Not A
100
150
250


Total
300
200
500