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Ten balls are picked without replacement from a box of 50 balls. All turn out to be red. A argues that the box contains only red balls. B argues that it contains 25 balls each of red and and some other colour. And in the pickings of the balls, the balls turned out to be red out of chance. C argues that there are 40 red balls and the rest of them are of some other colour. I have read statistical inference methods based on Bayes' theorem etc and to me, they appear to be making too many assumptions. I think the straight way of doing this is : Caclculate the probabilities of getting 10 red balls out of 10 pickings in the situation argued by A, B and C. And whichever situation gives the red balls with the most probability take that as the preferred possibility. Is this approach wrong?

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No, its not wrong. Its just not Bayesian. You are making an argument from Maximum Likelihood Inference vs Bayesian inference. Look up the Neyman-Pearson Lemma too see that you are in good company. Unless you know something about how the balls end up in the box, I would argue that you should argue from the most likely explanation. An alternative is to set up a hypothesis test for the number of red balls out of 50 and determine which null hypotheses will not be rejected for a given Type I error rate (e.g., 5%). This will generate a confidence interval, which is probably more useful than a single point estimate.

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