Box-$I$ contains $3$ cards bearing numbers $1, 2, 3$; Box-$II$ contains $5$ cards bearing numbers $1, 2, 3, 4, 5$ and Box-$III$ contains $7$ cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. One card is drawn at random from each of the boxes. If $x_i$ is the number on the card drawn from the $i^{\text{th}}$ box,$i=1, 2, 3$,then the probability that $x_1+x_2+x_3$ is odd is equal to

$12$ balls are distributed among $3$ boxes. The probability that the first box will contain exactly $3$ balls is:

$A$ ship is fitted with three engines $E_1, E_2$,and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4}$,and $\frac{1}{4}$. For the ship to be operational,at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_1, X_2$,and $X_3$ denote respectively the events that the engines $E_1, E_2$,and $E_3$ are functioning. Which of the following is (are) true?
$(A) P(X_1^c \mid X) = \frac{3}{16}$
$(B) P(\text{Exactly two engines are functioning} \mid X) = \frac{7}{8}$
$(C) P(X \mid X_2) = \frac{5}{16}$
$(D) P(X \mid X_1) = \frac{7}{16}$

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$,then which of the following is true?
$(A)$ $P(E)=\frac{4}{5}, P(F)=\frac{3}{5}$
$(B)$ $P(E)=\frac{1}{5}, P(F)=\frac{2}{5}$
$(C)$ $P(E)=\frac{2}{5}, P(F)=\frac{1}{5}$
$(D)$ $P(E)=\frac{3}{5}, P(F)=\frac{4}{5}$

If $A$ and $B$ are two independent events such that $P(\bar{A})=0.75$,$P(A \cup B)=0.65$ and $P(B)=x$,then find the value of $x$.

For independent events $A$ and $B$,if $P(A) = \frac{1}{2}$ and $P(A \cup B) = \frac{3}{5}$,then $P(B) =$ . . . . . . .