I went through binomial distribution and Chi-square test etc and got confused further. This question might be very basic and simple.
I have three states (Combination of two colors, both has equal chance) which are hidden (RedRed , RedBlue, BlueBlue) and each draw gives either Red or Blue. Red and Blue are independent. Now from N draws, I want to infer the likelihood of states with a given probability.
Example 1: n =20 ; Observed Red =12 & Blue= 10 then I want to find the likelihood of having RedRed , RedBlue & BlueBlue states. I guess it will be something like 0, 1, 0 respectively in this case.
But often n is small Example 2: n =4 ; Observed Red =1 & Blue= 3, Now how likely its BlueBlue, RedBlue & RedRed(unlikely )?.
I will really appreciate your answers. It wold be great if you comment any R function with this examples.
Thank you in advance
Since you only have two states, this is just a binomial prediction problem.
A common approach is to use a Beta-Binomial model,which allows you to specify the "prior" probabilities of different proportions of Blue vs Red and then use your sample to adjust this based on your observations. In fact, this Wiki article shows how this can be applied to your exact situation.
If you don't like Bayesian approaches, then you can also use a "prior free" likelihood-based approach. A simple prediction formula is given by Hinkley.