John & Jane are married. The probability that John watches a certain television show is .4. The probability that Jane watches the show is .5. The probability that John watches the show, given that Jane does, is .7.

1) find probability that both John & Jane watch the show.
2) Find the probability that Jane watches the show, given that John does.
3) Do John & Jane watch the show independently of each other?

A

P(Jane and John) = P(John | Jane) * P(Jane) = (0.7) * (0.5) = 0.35

B
P(Jane | John) = P(Jane and John) / P(John) = (0.35) / (0.4) = 0.875

C
P(Jane) * P(John) = (0.5) * (0.4) = 0.20

P(Jane and John) ≠ P(Jane) * P(John) = 0.35 ≠ 0.20

The two events are not independent do to the fact that P(Jane and John) ≠ P(Jane) X P(John). With that being said, John and Jane do not watch the show independently of each other.

1) To find the probability that both John and Jane watch the show, we can multiply their individual probabilities. So, the probability is 0.4 * 0.5 = 0.2.

2) To find the probability that Jane watches the show given that John does, we can use conditional probability. The probability is given as 0.7, which means that if John watches the show, there is a 70% chance that Jane will also watch it.

3) No, John and Jane do not watch the show independently of each other. The fact that the probability of John watching the show changes when we know that Jane is watching it indicates that their watching behaviors are not independent.

To find the probabilities, we can use the rules of conditional probability.

1) To find the probability that both John and Jane watch the show, we can use the formula for the intersection of two events:

P(John and Jane watch the show) = P(John watches the show) * P(Jane watches the show, given that John does)

P(John and Jane watch the show) = 0.4 * 0.7 = 0.28

So, the probability that both John and Jane watch the show is 0.28.

2) To find the probability that Jane watches the show, given that John does, we can use the formula for conditional probability:

P(Jane watches the show, given that John does) = P(John and Jane watch the show) / P(John watches the show)

P(Jane watches the show, given that John does) = 0.28 / 0.4 = 0.7

So, the probability that Jane watches the show, given that John does, is 0.7.

3) To determine whether John and Jane watch the show independently of each other, we need to compare the probability of both events happening together (watching the show) to the probability of the events happening separately (independently).

If the events are independent, then the probability of both events happening together should be equal to the product of the probabilities of each event happening separately.

P(John and Jane watch the show) = P(John watches the show) * P(Jane watches the show)

0.28 = 0.4 * 0.5

Since 0.28 is not equal to 0.4 * 0.5, we can conclude that John and Jane do not watch the show independently of each other.

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.