If A and B are two events such that [P(A)=(1)/(3), P(B)=(1)/(5)] and [P(A B)=(1)/(2)], the…

Question

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 :

Accepted Answer

$$P(A) = {1 \over 3},\,P(B) = {1 \over 5}$$ and $$P\left( {A \cup B} \right) = {1 \over 2}$$

$$\therefore$$ $$P\left( {A \cap B} \right) = {1 \over 3} + {1 \over 5} - {1 \over 2} = {1 \over {30}}$$

Now, $$P\left( {A|B'} \right) + P\left( {B|A'} \right) = {{P\left( {A \cap B'} \right)} \over {P\left( {B'} \right)}} + {{P\left( {B \cap A'} \right)} \over {P\left( {A'} \right)}}$$

$$ = {{{9 \over {30}}} \over {{4 \over 5}}} + {{{5 \over {30}}} \over {{2 \over 3}}} = {5 \over 8}$$

Solutions · 1