Compute $P(A | B),$ if $P(B)=0.5$ and $P(A \cap B)=0.32$.

$A$ bag contains $3$ red and $7$ black balls. Two balls are taken out at random,without replacement. If the first ball taken out is red,what is the probability that the second ball taken out is also red?

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

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

Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is $11$,given that $5$ appears on the first die?

Let $E$ and $F$ be events with $P(E)=\frac{3}{5}, P(F)=\frac{3}{10}$ and $P(E \cap F)=\frac{1}{5}$. Are $E$ and $F$ independent?