Probability of occurrence in a subset of a population If 100 individuals i.e. 2% of a population totalling 5000 people, have a certain characteristic, what is the probability of two or more persons within a subset of 10 persons, taken from that same population, having this characteristic?
Is there a particular formula, like a Z-test, for figuring this out?
 A: The exact method is to use the hypergeometric distribution. In terms
of balls in an urn, you have
and urn containing 5000 balls, of which 100 are red. You withdraw
10 balls without replacement, and want the probability of getting
at least two red balls, that is $P\{X \ge 2\}.$
Computation will be easier if you find $1 - P\{X \le 1\}.$
You can look up the standard formula for probabilities in a hypergeometric
distribution. I computed the answer using software as  0.01605178.
Because so few balls are taken from the urn the difference between
sampling with and without replacement is quite small. For sampling
with replacement, the number $X$ of red balls drawn has a binomial
distribution with $n = 10$ (draws) and $p = .02$ (probability any
one ball will be red. Then using standard formulas for
binomial probabilities we have $P\{X = 0\} = (.98)^{10}$ and 
$P\{X = 1\} = 10(.02)(.98)^9.$ Again here, you need to subtract
these two numbers from 1 to get your answer. Using software, I got
 0.01617764 for the answer. Notice that both answers amount to 0.016
when rounded to two decimal places.
Finally, you ask about using the normal distribution for this.
You could try to use an normal approximation, but with such a small
probability of getting the 'certain characteristic' on each draw,
the approximation would not be reliable.
