Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs $)=\frac{5}{9}$, is :
$\frac{1}{3}$
$\frac{2}{9}$
$\frac{4}{9}$
$\frac{5}{12}$
If $A$ and $B$ are two independent events such that $P(A) > 0.5,\,P(B) > 0.5,\,P(A \cap \bar B) = \frac{3}{{25}},\,P(\bar A \cap B) = \frac{8}{{25}}$ , then $P(A \cap B)$ is
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
$A$ and $B$ are events such that $P(A)=0.42$, $P(B)=0.48$ and $P(A$ and $B)=0.16 .$ Determine $P (A$ or $B).$
Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then
Two persons $A$ and $B$ throw a (fair)die (six-faced cube with faces numbered from $1$ to $6$ ) alternately, starting with $A$. The first person to get an outcome different from the previous one by the opponent wins. The probability that $B$ wins is