For the three events $A, B$ and $C, P$ (exactly one of the events $A$ or $B$ occurs) = $P$ (exactly one of the events $B$ or $C$ occurs)= $P$ (exactly one of the events $C$ or $A$ occurs)= $p$ and $P$ (all the three events occur simultaneously) $ = {p^2},$ where $0 < p < 1/2$. Then the probability of at least one of the three events $A, B$ and $C$ occurring is
$\frac{{3p + 2{p^2}}}{2}$
$\frac{{p + 3{p^2}}}{4}$
$\frac{{p + 3{p^2}}}{2}$
$\frac{{3p + 2{p^2}}}{4}$
In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is $0.8$ and the probability of passing the second examination is $0.7 .$ The probability of passing at least one of them is $0.95 .$ What is the probability of passing both ?
An experiment has $10$ equally likely outcomes. Let $\mathrm{A}$ and $\mathrm{B}$ be two non-empty events of the experiment. If $\mathrm{A}$ consists of $4$ outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
A card is drawn at random from a pack of cards. The probability of this card being a red or a queen is
If $A$ and $B$ are any two events, then $P(A \cup B) = $
If $A$ and $B$ are any two events, then the probability that exactly one of them occur is