To answer these questions, we can use the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred.
1) To find the probability that both John and Jane watch the show, we can simply multiply their individual probabilities.
So, P(John watches the show) = 0.4 and P(Jane watches the show) = 0.5. Therefore, the probability that both John and Jane watch the show is 0.4 * 0.5 = 0.2.
2) To find the probability that Jane watches the show, given that John does, we use the conditional probability formula: P(A | B) = P(A and B) / P(B).
Let A be the event that Jane watches the show, and B be the event that John watches the show. We want to find P(A | B), which is the probability that Jane watches the show given that John does.
We already know that P(John watches the show, given that Jane does) is 0.7. From part 1), we know that P(John and Jane watch the show) is 0.2. Therefore:
P(Jane watches the show, given that John does) = P(Jane and John watch the show) / P(John watches the show)
= 0.2 / 0.4 = 0.5.
3) To determine if John and Jane watch the show independently of each other, we need to compare the probability of both events occurring together (P(John and Jane watch the show)) with the product of their individual probabilities (P(John watches the show) * P(Jane watches the show)).
If the two probabilities are equal, then John and Jane watch the show independently. Otherwise, they are dependent on each other.
In our example, P(John and Jane watch the show) is 0.2, and P(John watches the show) * P(Jane watches the show) is 0.4 * 0.5 = 0.2.
Since these two probabilities are equal, it means that John and Jane watch the show independently of each other.