Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happen is $\frac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2}$. Then a value of $\frac{P(E)}{P(F)}$ is

Let the mean and variance of $7$ observations $2, 4, 10, x, 12, 14, y$ (where $x > y$) be $8$ and $16$ respectively. Two numbers are chosen from the set $\{1, 2, 3, x-4, y, 5\}$ one after another without replacement. Then the probability that the smaller number among the two chosen numbers is less than $4$ is:

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B)=\frac{1}{6}$ and $P(\bar{A} \cap \bar{B})=\frac{1}{3}$, then $P(A)$ is equal to (Here, $\bar{E}$ is the complement of the event $E$)

If $A$ and $B$ are two events of a random experiment such that $P(\bar{A})=\frac{2}{3}$, $P(B)=\frac{4}{15}$ and $P(A \cap \bar{B})=\frac{1}{5}$, then $\sqrt{195[P(B \mid(A \cup \bar{B}))+P(A \cup B)]} = $

An unbiased coin is tossed. If the result is a head,a pair of unbiased dice is rolled and the sum of the numbers on the two faces is noted. If the result is a tail,a card from a well-shuffled pack of eleven cards numbered $2, 3, 4, \dots, 12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is:

If $7$ dice are thrown simultaneously,then the probability that all six digits appear on the upper faces is equal to -