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

A radar system can detect an enemy plane in one out of ten consecutive scans.

The probability that it can detect an enemy plane atleast twice in four consecutive scans is

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive house get one sample is