If $A$ and $B$ are two independent events such that $P(A \cap B') = \frac{3}{25}$ and $P(A' \cap B) = \frac{8}{25}$,then $P(A) = $

Let $X$ be a random variable which takes values $1, 2, 3, 4$ such that $P(X=r) = K r^3$ where $r = 1, 2, 3, 4$. Then:

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails } | \text{ } B \text{ has failed})$.

Compute $P(A | B),$ if $P(B)=0.5$ and $P(A \cap B)=0.32$.

Two planes $I$ and $II$ drop bombs on a target. The probabilities of hitting the target by $I$ and $II$ are $0.3$ and $0.2$ respectively. The second plane drops the bomb only if the first plane fails to hit the target. What is the probability that the second plane hits the target?

$A$ fair die is rolled. Consider events $E=\{1,3,5\}, F=\{2,3\}$ and $G=\{2,3,4,5\}$. Find $P(E | F)$ and $P(F | E)$.