Question 1097609
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Given Info:


*[Tex \Large P(A) = \frac{1}{3}]


*[Tex \Large P(B) = \frac{1}{2}]


*[Tex \Large P(A \cup B) = \frac{3}{4}]


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What we want to solve for: *[Tex \Large P(A^c \cap B)]


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Before we can get to the answer, we need to find *[Tex \Large P(A \cap B)] first.


Use this equation:
*[Tex \Large P(A \cup B) = P(A) + P(B) - P(A \cap B)]


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Let *[Tex \Large x = P(A \cap B)]. Solve for x.


*[Tex \Large P(A \cup B) = P(A) + P(B) - P(A \cap B)]


*[Tex \Large \frac{3}{4} = \frac{1}{3} + \frac{1}{2} - x]


*[Tex \Large \frac{9}{12} = \frac{4}{12} + \frac{6}{12} - x]


*[Tex \Large \frac{9}{12} = \frac{4+6}{12} - x]


*[Tex \Large \frac{9}{12} = \frac{10}{12} - x]


*[Tex \Large \frac{9}{12} - \frac{10}{12}  = - x]


*[Tex \Large \frac{9-10}{12}  = - x]


*[Tex \Large \frac{-1}{12}  = - x]


*[Tex \Large x = \frac{1}{12}]


This means that *[Tex \Large P(A \cap B) = \frac{1}{12}]


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Use the previous result to compute *[Tex \Large P(A^c \cap B)]


*[Tex \Large P(A^c \cap B) = P(B) - P(A \cap B)]


*[Tex \Large P(A^c \cap B) = \frac{1}{2} - \frac{1}{12}]


*[Tex \Large P(A^c \cap B) = \frac{6}{12} - \frac{1}{12}]


*[Tex \Large P(A^c \cap B) = \frac{6-1}{12}]


*[Tex \Large P(A^c \cap B) = \frac{5}{12}]


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The final answer is *[Tex \Large \frac{5}{12}]


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