Let $S = \{1, 2, \dots, 20\}$. $A$ subset $B$ of $S$ is said to be "nice" if the sum of the elements of $B$ is $203$. Then the probability that a randomly chosen subset of $S$ is "nice" is

$A$ man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ points is $\frac{m}{n}$, where $\text{gcd}(m, n) = 1$, then $m + n$ is equal to:

$A$ computer program has two modules $X$ and $Y$ and errors in them occur independently. $X$ has an error with probability $0.1$ and $Y$ has an error with probability $0.3$. If an error in $X$ alone causes the program to crash with probability $0.5$, an error in $Y$ alone causes the program to crash with probability $0.7$, and an error in both $X$ and $Y$ causes the program to crash with probability $0.8$, then the probability that the program crashes is

$A, B, C$ 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:

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of $52$ playing cards by a person $B$. They win the game if $A$ gets a prime score as the sum of the numbers appearing on both the dice and $B$ gets a face card and a card having a prime number. Then the probability that both $A$ and $B$ win is:

$A$ bag $P$ contains $4$ red and $5$ black balls,another bag $Q$ contains $3$ red and $6$ black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$,then the probability that out of the three balls drawn two are black and one is red,is