Two sequences of coin flips:

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

THTHTHHTHTHHHHHHHTTTHTHHHTHHTT

Which was generated by fair coin, which by 2 headed? How did you decide?

You were a bit ranting in your previous response, but let me try to answer this. It is actually impossible to tell, given your sample, because we do not know the sample size. Given infinite samples, it is guaranteed that both of these runs are going to happen for a fair coin. On the other hand, if both are representative of full sample size, the later is clearly the fair coin because it more closely approximates 50/50.

Both sequences are equally improbable, but they're two outcomes among 2^30 others. What you want to be comparing is the sequence consisting entirely of heads to all the other potential outcomes. If it is a fair coin toss, the chance of getting all heads in a sequence of 30 is 1/1073741824. So is the chance of getting any one other sequence, but not any other sequence.

HHH.... is possible from a head fail fair coin, just improbable. If it happened, I would be astounded, but I wouldn’t feel like logic or the laws of the universe were being violated (I would look for a reality reset button though to pick out the improbable but not impossible occurrence).

assuming you want a bayesian approach? take a flat prior and look at the MAP? though you know its just going to be the observed rate. if you know you only have a fair coin or a double head coin just compare their likelihoods? what is the point you are trying to get at?

Solomonoff induction and universal probability.

P(all heads|double headed coin) = 1

P(all heads|fair coin) = 2^{-L}

Applying Bayes gets P(double headed coin|all heads) >> P(fair coin|all heads) for a long enough sequence.

thats what i suggested originally, essentially

how about something specific. how would you answer the question you proposed used bayesian methods?