$A, B, C$ 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:

For two events $A$ and $B$,if $P(A) = P(A|B) = \frac{1}{4}$ and $P(B|A) = \frac{1}{2}$,then which of the following is true?

$A$ and $B$ each select one number at random from the distinct numbers $1, 2, 3, \ldots, n$. The probability that the number selected by $A$ is less than the number selected by $B$ is $\frac{1009}{2019}$. The probability that the number selected by $B$ is the number immediately next to the number selected by $A$ is:

In a battery manufacturing factory, machines $P$, $Q$ and $R$ manufacture $20 \%$, $30 \%$ and $50 \%$ respectively of the total output. The chances that a defective battery is produced by these machines are $1 \%$, $1.5 \%$ and $2 \%$ respectively. If a battery is selected at random from the production, then the probability that it is defective is

$A$ 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 natural numbers are chosen at random from $1$ to $100$ and are multiplied. If $A$ is the event that the product is an even number and $B$ is the event that the product is divisible by $4$,then $P(A \cap \bar{B})=$