The maximum value of the variance of a Binomial distribution with parameters $n$ and $p$ is

$A$ student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n$,so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2}$,is:

In a random experiment,two dice are thrown and the sum of the numbers appeared on them is recorded. This experiment is repeated $9$ times. If the probability that a sum of $6$ appears at least once is $P_1$ and a sum of $8$ appears at least once is $P_2$,then $P_1 : P_2 =$

Let a biased coin be tossed $5$ times. If the probability of getting $4$ heads is equal to the probability of getting $5$ heads,then the probability of getting at most two heads is

For an initial screening of an entrance exam,a candidate is given $50$ problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$,then the probability that he is unable to solve less than $2$ problems is:

$A$ fair die with faces $1, 2, 3, 4, 5,$ and $6$ is tossed $4$ times. What is the probability that in all $4$ tosses,the value obtained is not less than $2$ and not greater than $5$ (in $/81$)?