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Suppose I have a hypothetical coin flip experiment where the probability $p$ of getting head in a single throw is unknown and can be anywhere such that $0 < p < 1$. I do have a random sample with $n$ flips and $x$ successes (heads). Flips are known to be independent of each other and the sample is assumed to be free of bias.

Obviously the best estimation for $p$ is $p = \frac{x}{n}$.

What I do want to find is a confidence interval $(p1, p2)$ so that $x$ is in the $2\sigma$ interval (or any arbitrary interval for that matter) of any $p \in (p1, p2)$ for $n$ tries.

How would I go about this? Is there an easy formula I can just plug $n$ and $x$ into?

I suppose I could do a binary search for $p1$ and $p2$ where I calculate the probability of getting at least/most $x$ heads in $n$ tries and repeat that until I find a $p1$/$p2$ that is "close enough" to the desired confidence interval, but that does not feel right. Is suppose I'm asking whether there is a closed form formula for this.

Sidenote: My statistics knowledge and research abilities have forsaken me on this. It feels like the most basic question in statistics, yet I could not find a satisfying answer. I'm probably lacking the keywords for this or misunderstood examples I came across.

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Would, e.g., the Clopper–Pearson method for confidence intervals work for you ?

The calculation is relatively accessible.

The following can be run in R, or at rdrr.io/snippets/.

The estimate is x/n, and the confidence interval is the Clopper–Pearson confidence interval for this proportion.

 x =  7
 n = 21
 binom.test(x, n, conf.level=0.95)


### number of successes = 7, number of trials = 21
###
### 95 percent confidence interval:
###    0.1458769 0.5696755
###
### sample estimates:
### probability of success 
###      0.3333333 

Some other potential methods are listed here, under Details: www.rdocumentation.org/packages/DescTools/versions/0.99.44/topics/BinomCI

"wald", "wilson", "wilsoncc", "agresti-coull", "jeffreys", "modified wilson", "modified jeffreys", "clopper-pearson", "arcsine", "logit", "witting", "pratt", "midp", "lik", "blaker"

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  • $\begingroup$ Thanks, that looks exactly like what I need. I integrated it in my excel worksheet using this , and the results seem plausible. $\endgroup$
    – LukeG
    Aug 9, 2022 at 12:40
  • $\begingroup$ While I have you here: The background of this question is a 15 sided unfair die. I modelled each side as a coin flip where either this side is rolled or any other side is rolled. Is this a sensible thing to do? Or is there a different approach that is more suitable for this situation? $\endgroup$
    – LukeG
    Aug 9, 2022 at 13:01
  • $\begingroup$ Wouldn't your observations be counts for each side of the die ? Like, if you roll it 1000 times, #1 comes up 70 times, and #2 comes up 50 times, and so on ? ... I think in this case you may want to look at multinomial confidence intervals for proportions. One method is the Sison-Glaz. For example, a 6-sided die is rolled 60 times. In R, observed = c(10, 10, 10, 6, 14, 10); library(DescTools); MultinomCI(observed, conf.level=0.95, method="sisonglaz") . Here all the the confidence intervals overlap 0.167, suggesting there's not good evidence that the die is not fair. $\endgroup$ Aug 9, 2022 at 16:38

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