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 $m : n =$

Consider $5$ independent Bernoulli trials,each with a probability of success $p$. If the probability of at least one failure is greater than or equal to $31/32$,then in which interval does $p$ lie?

If $X \sim B(4, p)$ and $2 P(X=3)=3 P(X=2)$,then the value of $p$ is:

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting a head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$,respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice,independently,and $\beta$ is the number of heads that appear when $C_2$ is tossed twice,independently. Then,the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal is:

If the sum of the mean and the variance of a Binomial distribution for $5$ trials is $1.8$,then the value of $p$ is

How many times must a man toss a fair coin so that the probability of having at least one head is more than $90 \%$?