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

Find $P(E | F)$ when two coins are tossed once,where $E$ is the event that no tail appears and $F$ is the event that no head appears.

Prove that if $E$ and $F$ are independent events,then so are the events $E$ and $F^{\prime}$.

Two dice are thrown simultaneously. If $A$ is the event of getting the sum of the numbers on two dice as greater than or equal to $8$ and $B$ is the event of getting a number less than or equal to $3$ on at least one of the dice. Then,$P(B / A) = $

In a city,$40\%$ of the people have brown hair,$25\%$ have brown eyes,and $15\%$ have both brown hair and brown eyes. If a person is selected at random from those having brown hair,what is the probability that they also have brown eyes?

If $A$ and $B$ are two non-mutually exclusive events such that $P(A \mid B) = P(B \mid A)$,then