# Calculate the number of trials to distinguish between 2 binomial distribution models

#### jimbo

##### New Member
Hello,

This sounds like a problem that has to be well known, but I can't find a good answer...

I have two competing models for the fraction of special objects ('n') in a sample ('N'). One model predicts that f= n/N = 0.15 of the objects are special, the other that f = 0.3 are special. Ie both models are binomial distributions in which there are N-n normal objects, and n special objects. For now let's assume these fractions are without an implicit uncertainty, but if it is easy to include, it would be useful to assume each is quoted as f +/- some amount.

If I could run an experiment where I looked at 'M' samples and the result was I found 'm' special objects with an associated uncertainty in my experimental measured fraction of special objects, how would I decide the minimum number of samples 'M' to look at in order to distinguish between the models at some given confidence level (say 3 sigma).

In other words, I want to determine at some confidence level which of the models is correct, and I want to do so as efficiently as possible allowing for the uncertainty in my experiment (let's say for example that I mistakenly identify some number of the objects, but that I can estimate the size of that uncertainty).

This would hopefully give me something as general as possible whereby I can look at the uncertainty in my measurements, the uncertainty in the models and the degree of confidence I require to happily distinguish between them and see the effects of changing these parameters.