Why are independent events modeled as a binomial random variable? I am currently studying for the first Actuarial exam and I am going over the practice questions SOA has on there website. The question I am having trouble with is:
"A company establishes a fund of 120 from with it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year are mutually independent. Calculate the Maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance."
On solutions it says this is a binomial random variable without actually explaining how they came to this conclusion. However I don't understand why? It seems to be a Central Limit Theorem question. Can somebody explain why it is binomial r.v.?
 A: The number of employees that reach "high performance level" follows a binomial distribution. There are $n = 20$ independent trials (employees) with probability $p = .02$ of success (reaching that level). You should recognize these conditions to be the binomial distribution when asking for the number of employees that reach this level. Thus, the probability the $k$ employees reach this level is
$$\binom{20}{k} (.02)^k (.98)^{20-k}.$$

If $X$ is the number of employees that reach high performance level, then $X$ follows distribution $\text{Binomial}(n = 20, p = .02)$. At the end of the year, the amount of funds $Y$ to pay is 
$$Y = CX.$$
Thus, you are asked to find $C$ such that
$$P(120-Y < 0) = P(120-CX<0) \leq .01$$
A: We are told that each event (an employee reaching high performance) is independent.  Thus the probability that all 20 reach high performance is:
$\underbrace{0.02 \cdot 0.02 \cdots 0.02}_{20}$.
The probability that exactly one reaches high performance can be obtained 20 different ways (for each of the employees), or ${20 \choose 1}$.  The probability exactly two is proportional to ${20 \choose 2}$, and so on.
This is the essence of a binomial distribution. 
A: We are looking at Bernoulli trials with out come 0 or C. Then, we are considering the sum (total amount pay to be paid) of these Bernoulli trial which is Binomial. 
Best of luck! 
