Question

Independence and conditional probability JR^(7) You've won a In an experiment, the probability that event A occurs is (8)/(9), the probability that event B occurs is (1)/(6), and the probability that event A occurs given that event B occurs is (4)/(9). Are A and B independent events?

Answer 1

Two events, A and B, are considered independent if the occurrence of one event does not affect the probability of the other event occurring. In mathematical terms, this means that:

P(A ∩ B) = P(A) * P(B)

Where:

- P(A ∩ B) is the probability of both events A and B occurring.

- P(A) is the probability of event A occurring.

- P(B) is the probability of event B occurring.

Let's calculate these probabilities based on the information given:

1. P(A) = 8/9

2. P(B) = 1/6

Now, we need to find P(A ∩ B), the probability of both events A and B occurring. We are given that the probability that event A occurs given that event B occurs is 4/9. In other words:

P(A | B) = 4/9

Using the definition of conditional probability:

P(A | B) = P(A ∩ B) / P(B)

We can rearrange this to find P(A ∩ B):

P(A ∩ B) = P(A | B) * P(B)

Plugging in the values:

P(A ∩ B) = (4/9) * (1/6) = 4/54 = 2/27

Now, let's check if A and B are independent:

P(A ∩ B) = P(A) * P(B)

2/27 = (8/9) * (1/6)

Now, simplify the right-hand side:

2/27 = 8/54

The two sides are not equal. Therefore, A and B are not independent events because the product of their individual probabilities does not equal the probability of both events occurring together (P(A ∩ B)). In this case, knowing that event B has occurred affects the probability of event A occurring, which means they are dependent events.

Explanation: