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I need to determine x given the following conditions.

A and B are two events of Ω.

P(not A) = 3x

P(B) = 1/2

P(A or B) = 9x

P(A and B) = 3x

Here's what I thought of (but I know is wrong):

P(A or B) = 9x P(A) + P(B) = 9x 1 - P(not A) + 1/2 = 9x 1 - 3x + 1/2 = 9x x = 1/8 

The answer is 0.1, but I can't get to it. I can't do the above because P(A or B) = P(A) + P(B) can only be done if P(A and B) = 0, and we are not told that here.

Any ideas? Thank you in advance.

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    Really @FrenzYDT.? Damn, I was pretty sure about that.2012-10-20

1 Answers 1

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You need to use that $A = (A \cup B)\setminus B \cup (A \cap B)$. Since $B \subset A \cup B$ and $(A \cup B)\setminus B \cap (A \cap B) = \emptyset$, you have $ P(A) = P(A \cup B) - P(B) + P(A \cap B) $ and get $ 1 - 3x = 9x - \frac{1}{2} + 3x $ which yields $x = \frac{1}{10}$.

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    Simpler than what I was preparing with conditional expectations; nice.2012-10-20