Find the binomial probability distribution whose mean is $3$ and variance is $2$.

If the sum and the product of the mean and variance of a binomial distribution are $24$ and $128$ respectively,then the probability of one or two successes is:

$A$ fair coin is tossed a fixed number of times. If the probability of getting $5$ heads is equal to the probability of getting $4$ heads, then the probability of getting $6$ heads is

The probability of securing a success in a trial is three times that of a failure. The probability of getting at least $4$ successes in $5$ trials is

Find the probability of getting $5$ exactly twice in $7$ throws of a die.

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: