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I think the formula should be (n+1)/(n+m+1) which should correspond to the mean of a binomial distribution with a uniform prior. So it's adding 1 to each count of observations.

This is probably the formula to memorise and check against.

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You mean (n+1)/(n+m+2) and yes, this is Laplace's rule of succession. It won't give you a confidence interval, but it gives you a posterior point estimate.

If you want a rough 95 % confidence interval without complicated maths, the Agresti–Coull interval is useful. It's computed as if the distribution was normal, but pretending there were two more successes and failures instead.

Yep, you're correct. It should be (n+1)/(n+m+2).

If you have access to a machine or lookup tables, you might as well plug in the values for the distribution Beta(1+n,1+m) which should correspond to the joint density.

(The formula above corresponds to the mean of this distribution, so it's probably right but I haven't work it through myself now ...)

See the Jeffrey posterior section here:

https://en.m.wikipedia.org/wiki/Binomial_proportion_confiden...

The blog post uses a non informative Jeffrey prior.

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