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I have a very basic understanding of statistics and probability. I just passed an intro college course on it. The problem is, there's one type of problem that I'd like to know how to do, and I don't have another statistics class, and it wasn't covered in this one. I think that there are other threads that answer this question, but the math is a bit over my head.

So what I'm wondering is how to calculate a percentile probability based on a recurring event. Let's say event A has a 2% probability of occurring. But what if we repeat event A x number of times? I understand the (and - multiplication) and (or - addition) rules. It seems that repeating this x number of times would be an or incident, since it gives a situation multiple opportunities to occur. But the problem with that is that with enough repetitions of x, we'd eventually, and rather quickly end up with P>1, which isn't how a percentage works.

For example, in our situation with a 2% likelihood, let's assume we repeat the situation 51 times. That's 51 different opportunities for the event to happen, so it falls into the or rule. If we add .02 together for all of this, we will end up with P>1. For one, this isn't how a percentile based calculation even works, since P cannot ever be greater than 1. Secondly, it seems unlikely that given 50 repetitions, something with a 2% likelihood would be certain to occur.

Basically, I know I'm doing this wrong, but I don't understand what about it is wrong, and I'd like to know the right way to do it. I've done a lot of digging, but the only things that I can find that closely relate to this do not explain it without breaking down whatever formula they're using. I'd really like to know how to do it, as well as why it works the way it does. I appreciate any answers.

Ryan Park
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    I think you are asking, "if an event occurs with probability $p$ and you observe $n$ independent trials, then what is the probability that the event is never observed?" That is $(1-p)^n$. If, instead, were asking "in the same scenario, what is the probability that the even is observed at least once?" that is the compliment, so the answer is $1-(1-p)^n$. – lulu Sep 12 '17 at 21:51
  • YES. Thank you so much, this is what I've been trying to figure out all day haha. – Ryan Park Sep 12 '17 at 22:02
  • It's a standard trick to work on the complimentary event. Doesn't always work, but when it does it is great. – lulu Sep 12 '17 at 22:02
  • Maybe one more thing of interest. Suppose you have a large number of trials and want to estimate the success probability $p.$ If $X$ is the observed number of successes in $n$ trials, then the estimate is $\hat p = X/n.$ Also, if $n$ is large enough a reasonably good 95% confidence interval for $\hat p$ is $\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/n}.$ This provides an indication how close $\hat p$ is to the population value $p.$ If you have a public opinion poll of 2500 people and 51% are in favor of Candidate A, then the CI is $.51 \pm .02$ so you still aren't sure who will win. – BruceET Sep 12 '17 at 22:32

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