Cards are drawn one-by-one without replacement from a well-shuffled pack of $52$ cards. The probability that a face card (jack, queen, or king) will appear for the first time on the third turn is equal to:

$A, B, C$ are mutually exclusive events such that $P(A) = \frac{3x+1}{3}$, $P(B) = \frac{1-x}{4}$, and $P(C) = \frac{1-2x}{2}$. Then the set of possible values of $x$ is:

The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$, respectively. If $A$ is selected as the captain, the probability that the team wins the tournament is $0.8$ and if $B$ is selected as the captain, the probability that the team wins the tournament is $0.7$. Then the probability, that the team wins the tournament, is:

Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}$,$P(Y \mid X)=\frac{1}{3}$,and $P(X \cap Y)=\frac{1}{6}$. Which of the following is (are) correct?
$(A)$ $P(X \cup Y)=\frac{2}{3}$
$(B)$ $X$ and $Y$ are independent
$(C)$ $X$ and $Y$ are not independent
$(D)$ $P(X^C \cap Y)=\frac{1}{3}$

If $A$ and $B$ are two events,then which of the following is $NOT$ true?

Four persons $A, B, C$ and $D$ throw an unbiased die, turn by turn in succession till one gets an even number and wins the game. What is the probability that $A$ wins the game if $A$ begins?