For three events $A, B$ and $C$ of a sample space, $P$ (exactly one of $A$ or $B$ occurs ) $=P$ (exactly one of $B$ or $C$ occurs) $=P($ exactly one of $C$ or $A$ occurs $)=\frac{1}{4}$. If probability of all the three events occurring simultaneously is $\frac{1}{16}$, then the probability that atleast one of the events occur 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

On every evening, a student either watches TV or reads a book. The probability of watching TV is $\frac{4}{5}$ If he watches TV, the probability that he will fall asleep is $\frac{3}{4}$ and it is $\frac{1}{4}$ when he reads a book. If the student is found to be asleep on an evening the probability that he watched the TV is

Let $X$ be the random variable taking values $1,2, \ldots n$ for a fixed positive integer $n$. If $P(X=k)=\frac{1}{n}$ for $1 \leq k \leq n$, then the variance of $X$ is