$A$ bag contains $7$ different black balls and $10$ different red balls. If balls are drawn one by one randomly until all black balls are drawn,what is the probability that this process is completed in the $12^{th}$ draw?

Two persons $A$ and $B$ are throwing an unbiased six-faced dice alternatively,with the condition that the person who throws $3$ first wins the game. If $A$ starts the game,then the probabilities of $A$ and $B$ winning the game are,respectively:

$A$ 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?
$(A) P(X_1^c \mid X) = \frac{3}{16}$
$(B) P(\text{Exactly two engines are functioning} \mid X) = \frac{7}{8}$
$(C) P(X \mid X_2) = \frac{5}{16}$
$(D) P(X \mid X_1) = \frac{7}{16}$

Two fair dice,each with faces numbered $1, 2, 3, 4, 5$ and $6$,are rolled together and the sum of the numbers on the faces is observed. This process is repeated until the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number,then the value of $14p$ is . . . . .

$A$ coin is tossed $2020$ times. The probability of getting head on $1947^{\text{th}}$ toss is

The probability that $A$ speaks truth is $75 \%$ and the probability that $B$ speaks truth is $80 \%$. The probability that they contradict each other when asked to speak on a fact is