...

In many problems, we need to determine a binomial probability (e.g. probability of a flood in a certain week of the year), or a proportion (e.g. the proportion of components that are made to a certain tolerance). To estimate the probability or proportion p you will have had some trials n, of which s were successes. This section describes three methods:

The crudest method, not recommended, but explained so you know why to avoid it.

Normal approximation to the binomial distribution method

Commonly used. It offers some improvement over the binomial method, but still cannot be applied when *s* = 0 or *n*, and gives incorrect results at extremes.

*Mid-p *cumulative confidence construction

The best method that works for all values of s and n. It is also closely aligned to Bayesian results. Useful when we don't know if the Binomial outcome is present.

...