$A$ and $B$ are two independent events of a random experiment and $P(A) > P(B)$. If the probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$,then the probability of the occurrence of $B$ is

If a $3$-digit number is randomly chosen,what is the probability that either the number itself or some permutation of the number (which is a $3$-digit number) is divisible by $4$ and $5$?

Let the probability of getting a head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64x^2 + 5Nx + 1 = 0$ has no real root is $\frac{p}{q}$,where $p$ and $q$ are co-prime,then $q - p$ is equal to

$A$ coin is tossed $8$ times. If the probability that exactly $4$ heads appear in the first six tosses and exactly $3$ heads appear in the last five tosses is $p$, then $96p$ is equal to ————

If $A, B$ and $C$ are three independent events of a random experiment such that $P(A \cap B^{c} \cap C^{c}) = \frac{1}{4}$,$P(A^{c} \cap B \cap C^{c}) = \frac{1}{8}$ and $P(A^{c} \cap B^{c} \cap C^{c}) = \frac{1}{4}$,then $P(A), P(B)$ and $P(C)$ are respectively

For three events $A, B$ and $C$,$P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \frac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \frac{1}{16}$. Then the probability that at least one of the events occurs is: