# Probability that any $k$ of $n$ do not get an event

The probability of an independent event happening to a thing is $$0 \leq p \leq 1$$. There are $$0 \leq n$$ discrete things. What is the probability $$P$$ that any combination of $$0 \leq k \leq n$$ do not experience the event?

I can work this out by hand for any two of three things ($$n = 3, k = 2$$). Let's say the three things are $$A, B,$$ and $$C:$$

| Good scenario     | Probability     |
|-------------------|-----------------|
| A and B and not C | p * p * (1 - p) |
| A and C and not B | p * (1 - p) * p |
| B and C and not A | (1 - p) * p * p |


So $$P(\text{Good scenario}) = p^2 \cdot (1-p)$$, and there are 3 possible good scenarios.

So what's the probability of any of these scenarios happening? I think it's the probability of not(every good scenario fails)... $$1 - (1 - P(\text{Good scenario}))^3$$

But how do I generalize this? Is the following correct?

$$P(\text{Good scenario}) = p^k \cdot (1 - p)^{n - k}$$ $$P = 1 - (1 - P(\text{Good scenario}))^{n \choose k}$$

• For any two of three things, you have successfully shown that each one individually happens with probability $p^2 (1-p)$ but because all of these consist of a "Good scenario" we must add these together to get a probability of a Good scenario to be $3p^2 (1-p)$ – WaveX Dec 31 '19 at 16:58
• The phrasing is not clear. Do you mean "having specified a set of $k$ things, what is the probability that those $k$ $\textit {and no other}$ things do not experience the event?" Or do you mean "what is the probability that the set of things which do not experience the event has exactly $k$ elements?" or do you mean something else? – lulu Dec 31 '19 at 17:03

$$\mathrm{Pr}[k\textrm{ out of }n\textrm{ objects experience a probability-}p\textrm{ event}]\equiv \mathrm{Pr}(k; n, p)=\binom{n}{k}p^k(1-p)^{n-k}$$
where $$\binom{n}{k}$$ is the number of ways to choose $$k$$ out of the $$n$$ objects (i.e., the number of "good scenarios", and $$p^k(1-p)^{n-k}$$ is the probability of each one of these "good scenarios" happening.
Note: the title of your question says "Probability that any $$k$$ of $$n$$ do not get an event", but in your example you calculate the probability that any $$k$$ of $$n$$ do get the event. It is not clear to me which one you are looking for. My answer gives the probability of getting the event: if you want the probability of not getting it, just replace $$p$$ by $$1-p$$.