Probability of island having 8 people born with disease, estimate? The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that particular disease? We assume that all $10000$ events of being born with that particular disease are mutually independent.

This random variable has the binomial distribution. Hence, the probability that an island with $10000$ inhabitants has precisely $8$ people born with that particular disease is given by
$$
{10000\choose 8}\biggl(\frac1{1200}\biggr)^8\biggl(1-\frac1{1200}\biggr)^{(10000-8)}\approx0.1387.
$$
