How many times must a fair coin be tossed so that the probability of getting at least one head is at least $0.8$?

If there are $n$ independent trials,$p$ and $q$ are the probability of success and failure respectively,then what is the probability of exactly $r$ successes?

Suppose $X$ has a binomial distribution $B(6, 1/2)$. Show that $X=3$ is the most likely outcome.
(Hint: $P(X=3)$ is the maximum among all $P(x_i)$,where $x_i = 0, 1, 2, 3, 4, 5, 6$)

If a random variable $X$ follows the binomial distribution with parameters $n=5, p$ and $P(X=2)=9 P(X=3)$,then $p$ is equal to

The probability of a shooter hitting a target is $\frac{3}{4}$. What is the minimum number of times he/she must fire so that the probability of hitting the target at least once is greater than $0.99$?

An urn contains $25$ balls of which $10$ balls bear a mark $'X'$ and the remaining $15$ bear a mark $'Y'$. $A$ ball is drawn at random from the urn,its mark is noted down and it is replaced. If $6$ balls are drawn in this way,find the probability that all will bear $'X'$ mark.