$A$ multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is

$A$ coin is tossed $7$ times. What is the probability that a person who calls 'heads' every time wins more often?

In a binomial distribution,the difference between the mean and standard deviation is $3$ and the difference between their squares is $21$,then $P(x=1) : P(x=2) =$

For a binomial variate $X \sim B(n, p)$,the difference between the mean and variance is $1$ and the difference between their squares is $11$. If the probability $P(X=2) = m\left(\frac{5}{6}\right)^n$ and $n=36$,then $m : n =$

$A$ fair coin is tossed $K$ times such that the probability of getting $4$ heads is equal to that of getting $6$ heads. If the probability is maximum for getting $r$ heads,then $r=$

$A$ bag contains $4$ red and $3$ black balls. One ball is drawn and then replaced in the bag and the process is repeated. Let $X$ denote the number of times a black ball is drawn in $3$ draws. Assuming that at each draw each ball is equally likely to be selected,the probability distribution of $X$ is given by: