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Mathematics Question on Probability

If P(A)=112,P(B)=512, P\left(A\right)=\frac{1}{12}, P\left(B\right)=\frac{5}{12}, and P(B∣A)=115P\left(B|A\right) =\frac{1}{15} then p(A∪B)p\left(A\cup B\right) is equal to

A

89180\frac{89}{180}

B

90180\frac{90}{180}

C

91180\frac{91}{180}

D

92180\frac{92}{180}

Answer

89180\frac{89}{180}

Explanation

Solution

Given
P(A)=112,P(B)=512,P(B/A)=115P\left(A\right)=\frac{1}{12},P\left(B\right)=\frac{5}{12}, P\left(B/A\right)=\frac{1}{15}
We know that P(B/A)=P(A∩B)P(A)P\left(B/A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}
⇒115=P(A∩B)1/12\Rightarrow \frac{1}{15}=\frac{P\left(A\cap B\right)}{1/12}
⇒P(A∩B)=115×12=1180\Rightarrow P\left(A\cap B\right)=\frac{1}{15\times12}=\frac{1}{180}
But,
P(A∪B)=P(A)+P(B)−P(A∩B)P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)
⇒P(A∪B)=112+512−1180=89180\Rightarrow P\left(A\cup B\right)=\frac{1}{12}+\frac{5}{12}-\frac{1}{180}=\frac{89}{180}