So I have this problem before me:
Jane is taking two books on her holiday vacation. With probability
0.5, she will like the first book; with probability 0.4 she will like
the second book; and with probability 0.3 she will like both books.
What is the probability she likes neither?
Now the solution is explained in the book I have, but what I don't understand is how it's possible that Jane will like both books with probability 0.3. Given that she will like the first with 0.5 probability and the second with 0.4, should the probability of Jane liking both be 0.5 * 0.4 = 0.2?
Best Answer
The product rule only holds for independent outcomes, but the problem does not state that Jane's opinions about the two books are independent. Perhaps they're written by the same author, in which case liking one makes it more likely that you'll also like the other.
In the dependent case, there is no necessary relation between the four probabilities P(likes only A), P(likes only B), P(likes both), P(likes none) -- except that their sum has to be 1. The problem gives you P(likes both), but you need to treat two of the others. You then have the information
and these two equations allow you to determine the remaining probabilities.