If the probability that an individual will suffer a reaction from an injection of a drug is $0.001$, then the probability that out of $2000$ individuals having that injection, more than $2$ individuals will suffer a reaction, 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?

India plays two matches each against Australia and West Indies. The probabilities of India scoring $0, 1,$ and $2$ points in a match are $0.45, 0.05,$ and $0.50$ respectively. Assuming the results are independent,what is the probability that India scores at least $7$ points?

$A$ boy tosses a fair coin $3$ times. If he gets $₹ 2x$ for $x$ heads,then his expected gain equals to $₹........$

$A$ card is drawn from a pack of $52$ playing cards. The card is replaced and the pack is shuffled. If this process is repeated $6$ times,what is the probability that $2$ hearts,$2$ diamonds,and $2$ black cards are drawn?

If a discrete random variable $X$ has the probability distribution $P(X=x) = k \frac{2^{2x+1}}{(2x+1)!}$ for $x = 0, 1, 2, \ldots, \infty$,then $k =$