Zero-two events in 30 observations. What is the quickest estimation of the upper limit of event frequency ?
A simple binomial p2(1-p)n-2 helps visualizing the distribution of the unknown frequency (p), from near zero to about 1/3. The mode is 1/15 (about 7%) and is not centered.
The 0-0.95 interval is the solution of this equation. But is this equation really meaningful ?
Such a calculation is nicely wrapped, but it starts from the hidden assumption that all frequencies are equally probable (we all live in a Bayesian world),
And the 0-0.95 interval is only a mental construction designed to translate a continuous range of value in a more manageable black and white picture of reality. This is not unsane but legitimates approximations. From a "realistic" perspective, the difference is small potatoes.
The rule of three is a simple approximation when no events are observed but also quite weird since it gives the same confidence intervals for possible and impossible events. What has occured at least twice are more likely to be observed again (about three times the observed frequency).
Sources and references
JAMA , 1983, Vol 249, n°13, p 1743-1745
BMJ, 1995, 311, p 619-620
Intuitive Biostatistics, 1995
Ann Génét, 1996, vol 39 n°3, p 133-138