I have in total $n$ people of which $b$ are bad people and I want to randomly select $v = 5$ people. I want to calculate the probability to choose at least $3$ bad people.
I calculated it as follows:
$$P = \frac{{b\choose 3}*{n-b\choose 2}+{b\choose 4}*{n-b\choose 1}+{b\choose v}}{{n \choose v}} $$
where:
${b\choose 3}*{n-b\choose 2} $ are the ways to choose $3$ bad people and then $2$ good ones,
${b\choose 4}*{n-b\choose 1} $ are the ways to choose $4$ bad people and then a good one,
and finally ${b\choose v}$ are the ways to choose only bad people.
Is this correct?
Your answer is good, I could understand it, but it is suggested either you write $v$ as $5$ and not have $v$ in your equation, or you have $v$ in your equation, and no $2, 3$, etc. That is for $v \le b$:
$$P = \frac{\Sigma_{x=3}^v\bigg({b\choose x}\cdot{n-b\choose v-x}\bigg)}{{n \choose v}} $$