Question

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) = .

Answer 1

Since P(G ∩ X) ≠ P(G) · P(X), the events of using Brand X and giving up doing laundry are not independent. In this case, a user of Brand X detergent is more likely to give up doing laundry than a randomly chosen person.

Explanation:

To determine whether the events of using Brand X and giving up doing laundry are independent, we need to compare the probability of the intersection of the two events, P(G ∩ X), with the product of the probabilities of each event, P(G) and P(X).

Given information:

- P(G) = 0.06 (probability of giving up doing laundry)

- P(X) = 0.70 (probability of using Brand X)

- P(G ∩ X) = 0.07 (probability of using Brand X and giving up doing laundry)

To test independence, we need to check if P(G ∩ X) is equal to P(G) · P(X):

- P(G) · P(X) = 0.06 · 0.70 = 0.042

Now, we compare P(G ∩ X) to P(G) · P(X):

- P(G ∩ X) = 0.07

- P(G) · P(X) = 0.042

Since P(G ∩ X) ≠ P(G) · P(X), the events of using Brand X and giving up doing laundry are not independent. In this case, a user of Brand X detergent is more likely to give up doing laundry than a randomly chosen person.