Question

There is a 20% probability that a person inoculated with a particular vaccine will get the disease anyway. A county healthy office inoculates 83 people. What is the probability that exactly 10 of them will get the disease at some point in their lives?

Answer 1

Let $X$ be a random variable denoting the number of inoculated people in a population of size $n$ that contract this disease. $X$ is said to be binomially distributed with probability $p=0.2$, or $X\sim\mathcal B(83,0.2)$.

So the probability that exactly 10 of 83 people get the disease, despite inoculation against it, is

$\mathbb P(X=10)=\dbinom{83}{10}0.2^{10}(1-0.2)^{83-10}\approx0.0209842$