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?
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.