JEE MAIN - Mathematics (2022 - 25th July Evening Shift - No. 6)
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 :
$$\frac{3}{4}$$
$$\frac{5}{8}$$
$$\frac{5}{4}$$
$$\frac{7}{8}$$
Explanation
$$P(A) = {1 \over 3},\,P(B) = {1 \over 5}$$ and $$P\left( {A \cup B} \right) = {1 \over 2}$$
$$\therefore$$ $$P\left( {A \cap B} \right) = {1 \over 3} + {1 \over 5} - {1 \over 2} = {1 \over {30}}$$
_25th_July_Evening_Shift_en_6_1.png)
Now, $$P\left( {A|B'} \right) + P\left( {B|A'} \right) = {{P\left( {A \cap B'} \right)} \over {P\left( {B'} \right)}} + {{P\left( {B \cap A'} \right)} \over {P\left( {A'} \right)}}$$
$$ = {{{9 \over {30}}} \over {{4 \over 5}}} + {{{5 \over {30}}} \over {{2 \over 3}}} = {5 \over 8}$$
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