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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.

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