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  4. If A and B are two events such that P ( A )=(1/3), P ( B )=(1/5) and P ( A ∪ B )=(1/2), then P ( A mid B prime)+ P ( B mid A prime) is equal to

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Q. If $A$ and $B$ are two events such that $P ( A )=\frac{1}{3}, P ( B )=\frac{1}{5}$ and $P ( A \cup B )=\frac{1}{2}$, then $P \left( A \mid B ^{\prime}\right)+ P \left( B \mid A ^{\prime}\right)$ is equal to

JEE MainJEE Main 2022Probability - Part 2

Solution:

$ P ( A \cup B )= P ( A )+ P ( B )- P ( A \cap B ) $
$ \frac{1}{2}=\frac{1}{3}+\frac{1}{5}- P ( A \cap B ) $
$ P ( A \cap B )=\frac{1}{30}$
$ P \left(\frac{ A }{\overline{ B }}\right)+ P \left(\frac{ B }{\overline{ A }}\right)=\frac{ P ( A \cap \overline{ B })}{ P (\overline{ B })}+\frac{ P ( B \cap \overline{ A })}{ P (\overline{ A })}$
$ =\frac{ P ( A )- P \left( A \cap B ^{\prime}\right)}{1- P ( B )}+\frac{ P ( B )- P ( A \cap B )}{1- P ( A )} $
$ =\frac{\frac{1}{3}-\frac{1}{30}}{\frac{4}{5}}+\frac{\frac{1}{5}-\frac{1}{30}}{\frac{2}{3}}=\frac{5}{8}$