$A$ die is tossed thrice. $A$ success is defined as getting $1$ or $6$ on a toss. Find the mean and the variance of the number of successes.

The probability that a bulb produced by a factory will fuse after $150$ days of use is $0.05$. Find the probability that out of $5$ such bulbs,none will fuse after $150$ days.

The probability that a bulb produced by a factory will fuse after $150$ days of use is $0.05$. Find the probability that out of $5$ such bulbs,more than one will fuse after $150$ days.

If the mean and variance of a binomial variable $X$ are $2$ and $1$ respectively,then $P(X>1)=$

As a business strategy, $20 \%$ of the new internet service subscribers selected randomly receive a special promotion. If a group of $5$ such subscribers signs for the service, then the probability that at least two of them get the special promotion 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: