If two events $A$ and $B$ are such that $P\,(A + B) = \frac{5}{6},$ $P\,(AB) = \frac{1}{3}\,$ and $P\,(\bar A) = \frac{1}{2},$ then the events $A$ and $B$ are

Four persons can hit a target correctly with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and $\frac {1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is

Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined $P ( A )=0.5$,  $ P ( B )=0.7$,  $P ( A \cap B )=0.6$

For any two independent events ${E_1}$ and ${E_2},$ $P\,\{ ({E_1} \cup {E_2}) \cap ({\bar E_1} \cap {\bar E_2})\} $ is

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

Two aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $l$ and $II$ scoring a hit correctlyare $0.3$ and $0.2,$ respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is