The probability that an individual suffers a bad reaction from an injection is $0.001$. The probability that out of $2000$ individuals exactly three will suffer a bad reaction is:

If $X$ is a random variable with probability distribution $P(X=k) = \frac{(2k+3)c}{3^k}$,$k=0, 1, 2, \ldots, \infty$,then $P(X=3) =$

In a book consisting of $600$ pages,there are $60$ typographical errors. The probability that a randomly chosen page will contain at most two errors is

If the probability density function of a continuous random variable $X$ is $f(x) = \frac{x^3}{3}$ for $-1 < x < 2$,and $f(x) = 0$ otherwise,then the cumulative distribution function $F(x)$ for $-1 < x < 2$ is:

Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values $x$ has the following form,where $k$ is some unknown constant.
$P(X=x) = \begin{cases} 0.1, & \text{if } x=0 \\ kx, & \text{if } x=1 \text{ or } 2 \\ k(5-x), & \text{if } x=3 \text{ or } 4 \\ 0, & \text{otherwise} \end{cases}$
What is the probability that you study at least two hours? Exactly two hours? At most two hours?

It is known that a box of $8$ batteries contains $3$ defective pieces and a person randomly selects two batteries from the box. If $X$ is the number of defective batteries selected,then $P(X \leq 1) = $