What error bar to use when p heads in a sample is equal to 1

Question

I essentially have an experiment where a coin with an unknown $p(heads)$ is flipped $N_{total}$ times. I want to plot $p(heads)$ along with the associated 95% CI. I was planning on plotting $p = N_{heads}/N_{total}$ with error bars derived from the inverse binomial distribution evaluated at 0.975 and 0.025 with parameters $p$ and $N_{total}$. The problem is that in my experiment $N_{heads}$ is equal to $N_{total}$ which gives me a CI zero height suggesting that I know the value of $p(heads)$ exactly, but that seems unlikely.

Is there a better way to calculate the CI? I was thinking I could use the binomial distribution to find the value of $p_{heads}$ that has a probability of having $N_{heads} = N_{total}$ equal to 0.975, but this seems like the exact opposite of what CIs are supposed to tell you.

Answer 1

Short answer, but this should work... you can estimate a CI for p(heads) using an Agresti-Coull interval:

https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Agresti.E2.80.93Coull_interval

This interval calculation method has 'good' coverage properties: http://www.stat.ufl.edu/~aa/articles/agresti_coull_1998.pdf