If I compute the observed probability of an event like so:
observed probability = positive outcome/total number of outcomes
How can I compute confidence intervals for the observed probability?
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1$\begingroup$ Welcome to CV, Bob. As you might imagine, we have many posts about this because such frequency estimates are really basic. You will find a range of solutions by searching for binomial confidence intervals. I have narrowed the search to focus on posts that mention one of the best, the Clopper-Pearson interval. $\endgroup$whuber– whuber ♦2025-06-10 17:44:14 +00:00Commented Jun 10 at 17:44
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The confidence interval of a probability (or proportion) is usually covered in introductory statistics class (though they usually only teach one option and point out that it does not work in all cases).
There are actually several formulas, each with different properties and conditions when it is appropriate. This Wikipedia article: https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval gives details on several options with discussions of their advantages and disadvantages.
In R you can just do:
binom.test(x, n)
and the output will include a 95% confidence interval (optionally you can choose a different coverage). There should be similarly simple commands in other languages like Python and Julia and there are probably online calculators as well. But if you use any of these quick and easy approaches, you are letting someone else make decisions and assumptions for you.
