If $E$ and $F$ are independent events such that $0 < P(E) < 1$ and $0 < P(F) < 1,$ then

If $A$ and $B$ are independent events and $P(A) = p, P(B) = 2p$,and $P(\text{exactly one from } A \text{ and } B) = \frac{5}{9}$,then find the value of $p$.

Let the mean and variance of $7$ observations $2, 4, 10, x, 12, 14, y$ (where $x > y$) be $8$ and $16$ respectively. Two numbers are chosen from the set $\{1, 2, 3, x-4, y, 5\}$ one after another without replacement. Then the probability that the smaller number among the two chosen numbers is less than $4$ is:

Let $A, B,$ and $C$ be $3$ independent events such that $P(A) = 1/3, P(B) = 1/2,$ and $P(C) = 1/4$. Find the probability that exactly $2$ of the $3$ events occur.

If $A$ and $B$ are two independent events such that $P(A) > 0.5$,$P(B) > 0.5$,$P(A \cap \bar{B}) = \frac{3}{25}$,and $P(\bar{A} \cap B) = \frac{8}{25}$,then $P(A \cap B)$ is:

$P$ speaks truth in $70\%$ of the cases and $Q$ in $80\%$ of the cases. In what percent of cases are they likely to agree in stating the same fact (in $\%$)?