Consider I have a jar with marbles labeled 0 and 1 in it. They're not well mixed so the possibility of obtaining a sample sized 1000 with mean 0.6 and another sample sized 1000 with mean 0.4 is not so unlikely as random selection hypothesis would suggest. How could I determine the mean of this population, other than expanding the sample size indefinitely? And how could I quantify my confidence level in this estimate?
P.S. I'm not familiar with Bayesian statistics, but I'm open to new information if necessary.
P.P.S. The jar model is not really relevant. Just consider I have a black box that gives random numbers.