Confidence interval for number of an event happening with a known probability

Question

As an example, my problem is something similar to this: Lets say that you open 10 chests and each chest has a 30% chance of having an item. Calculate the confidence interval for the number of items found. 95% confidence, use Student t distribution if possible.

I was using the formula $s(p) = \sqrt{p(1-p)}$ for estimating the standard error of the probability p and the Student t distribution for the confidence interval.

The number of items found is I=n*p; where n is the number of chests opened (n=10) and p is the probability of the chest having an item (p=0.3).

I would need something like: if I open 10 chests, there's a 95% probability that the number of items found is between 2 and 7.

Answer 1

You only need the variance of $\hat p$ if that is your target of inference, in this case you know the proportion and want to predict the number of 'successes' $X$ instead. You can draw this directly from the binomial point mass function, which looks like this for $n=10$ and $p=0.3$:

You can get the 95% central density of this distribution as its $(0.025, 0.975)$ quantiles, which in this case is $(0, 6)$. As an aside, even for inference on $p$ the normal approximation with which you started is almost always the worst option (its main advantage being ease of computation), especially for such small sample sizes.

I used the following R code to get to these numbers:

n <- 10
p <- 0.3

pmf <- dbinom(0:n, n, p)
ci <- qbinom(c(.025, .975), n, p)
> [1] 0 6