A market survey shows that 70% of the population used Brand X laundry detergent last year, 6% of the population gave up doing its laundry last year, and 7% of the population used Brand X and then gave up doing laundry last year. Are the events of using Brand X and giving up doing laundry independent? Is a user of Brand X detergent more or less likely to give up doing laundry than a randomly chosen person? Step 1
First, we need to test whether the two events are independent.
Use X to denote the event described by "A person used Brand X," and G to describe the event "A person gave up doing laundry."
Recall that the two events are independent if and only if the probability of
G ∩ X
is equal to the product of the probabilities of X and of G. That is, if and only if
P(G ∩ X) = P(G) · P(X).
To answer the question, calculate P(G), P(X), and
P(G ∩ X)
and then compare
P(G ∩ X)
to
P(G) · P(X).
Because 6% of the population gave up doing laundry, the probability that someone quit doing laundry is
P(G) = 0.06.
Similarly, 70% of the population used Brand X, so the probability that someone was a Brand X user is
P(X) = .
Furthermore, 7% of the population used Brand X and then gave up doing laundry, so the probability that someone was initially a Brand X user and then quit doing laundry is
P(G ∩ X) = .