Two cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. The mean of the number of aces is:

$A$ die is thrown four times. The probability of getting a perfect square in at least one throw is

$A$ bag contains $4$ red and $3$ black balls. One ball is drawn and then replaced in the bag and the process is repeated. Let $X$ denote the number of times a black ball is drawn in $3$ draws. Assuming that at each draw each ball is equally likely to be selected,then the probability distribution of $X$ is given by:

There are $6$ boxes labelled $B_1, B_2, \ldots, B_6$. In each trial,two fair dice $D_1, D_2$ are thrown. If $D_1$ shows $j$ and $D_2$ shows $k$,then $j$ balls are put into the box $B_k$. After $n$ trials,what is the probability that $B_1$ contains at most one ball?

Let $X \sim B(n, p)$. If $E(X)=5$ and $\operatorname{Var}(X)=2.5$,then find $P(X < 1)$.

In an examination,there are $10$ true-false type questions. Out of $10$,a student can guess the answer of $4$ questions correctly with probability $\frac{3}{4}$ and the remaining $6$ questions correctly with probability $\frac{1}{4}$. If the probability that the student guesses the answers of exactly $8$ questions correctly out of $10$ is $\frac{27 k}{4^{10}}$,then $k$ is equal to