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Say I have an event that has a probability of occurring $p$, and there are $n$ independent experiments that take place where the event is possible in the experiment.

1) What is the probability the event happens $m$ times?

By extension:

2) What is the probability the event happens at least $m$ times?

There are various questions answered here like "Probability of an unlikely event repeated many times" which answer the question, "How likely is it for an event to be repeated consecutively $m$ times?" This is not my question.

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This is the binomial distribution $n\choose m$ $p^m(1-p)^{n-m}$ or something like that. The second question is a tail probability, a sum of these binomial probabilities over the desired range.

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That is simply the binomial distribution. Given that the probability event "A" happens is p, then the probability it happens exactly m out of n times is $\frac{n!}{m!(n-m)!}p^m(1-p)^{n-m}$. The probability it happens at least m times out of n is a sum of those: $\sum_{i=m}^n \frac{n!}{i!(n-i)!}p^i(1-p)^{n-i}$

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