Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. What is the probability that all the five cards are spades?

If the mean and variance of a binomial distribution are $\frac{4}{3}$ and $\frac{10}{9}$ respectively, then $P(X \geq 6)=$

If a fair coin is tossed $10$ times,find the probability of exactly six heads.

Ten eggs are drawn successively with replacement from a lot containing $10 \%$ defective eggs. Find the probability that there is at least one defective egg.

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:

$A$ fair coin is tossed $99$ times. If $X$ is the number of times head occurs,then $P[X=r]$ is maximum when $r=$