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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?

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You're assuming that Jane's opinions about the two books are independent, but the problem does not state that. (Perhaps they're written by the same author, in which case liking one makes it more likely that you'll also like the other). –  Henning Makholm Sep 20 '11 at 17:48
    
Ah, okay that makes more sense now. I couldn't form the model in my head for how it could possibly be. Thank you! –  Ceasar Bautista Sep 20 '11 at 17:52
    
@mixedmath, mostly because I don't consider it a complete answer as stated. It is more of a hint than actual information. I have extended it to a minimal answer now such that the question will not look unanswered. –  Henning Makholm Sep 20 '11 at 21:16
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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

  • P(likes A) = P(likes only A) + P(likes both) = 0.5
  • P(likes B) = P(likes only B) + P(likes both) = 0.4

and these two equations allow you to determine the remaining probabilities.

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