# Probability that at least a certain number of people is chosen

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?

• That's all good! Feb 19, 2018 at 16:42
• In a couple of places you could replace $v$ with its known value, $5$. Feb 19, 2018 at 16:44

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 $v \le b$:
$$P = \frac{\Sigma_{x=3}^v\bigg({b\choose x}\cdot{n-b\choose v-x}\bigg)}{{n \choose v}}$$