$A$ black and a red dice are rolled. Find the conditional probability of obtaining a sum greater than $9,$ given that the black die resulted in a $5$.

An urn contains $5$ red and $2$ green balls. $A$ ball is drawn at random from the urn. If the drawn ball is green,then a red ball is added to the urn and if the drawn ball is red,then a green ball is added to the urn; the original ball is not returned to the urn. Now,a second ball is drawn at random from it. The probability that the second ball is red,is

If $A$ and $B$ are independent events such that $P(A \cap B') = \frac{3}{25}$ and $P(A' \cap B) = \frac{8}{25}$,then $P(A) =$

If $P(A)=0.8, P(B)=0.5$ and $P(B | A)=0.4,$ find $P(A \cap B).$

For two events $A$ and $B$,if $P(A) = P\left( \frac{A}{B} \right) = \frac{1}{4}$ and $P\left( \frac{B}{A} \right) = \frac{1}{2},$ then:

If $A$ and $B$ are any two events such that $P(A) + P(B) - P(A \cap B) = P(A)$, then $\dots \dots \dots$