If $A$ and $B$ are two independent events such that $P(B)=\frac{2}{7}$ and $P(A \cup B^c)=0.8$,then $P(A)$ is equal to:

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

If $P(E)=0.8, P(F)=0.5$ and $P(F \mid E)=0.4$ then,$P(E \mid F)=$ . . . . . . .

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:

$A$ pair of dice is thrown. If $5$ appears on at least one of the dice,then the probability that the sum is $10$ or greater is:

$A$ candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ if he passes the preceding one,and $\frac{p}{2}$ if he fails the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is: