Which urn? A Bayesian posterior from two draws

Question

Two urns sit on a table. Urn A holds 3 red and 1 blue ball; Urn B holds 2 red and 2 blue balls. You pick an urn at random (equal chance), then draw a ball, note its color, replace it, and draw again. Both draws are red.

Using Bayes' rule, what is the probability that you were drawing from Urn A?

Accepted Answer

Intuition first. Urn A is redder than Urn B, so two red draws lean toward Urn A, but not overwhelmingly since Urn B still has plenty of red. Bayes' rule quantifies the lean by weighing how likely two reds are under each urn, from the even 50/50 prior. The recipe. Multiply each urn's prior by the chance it produces two reds, then normalize. Chance of red on one draw: Urn A gives 3/4 = 0.75, Urn B gives 2/4 = 0.5. With replacement, two reds: P(RR\mid A) = 0.75^2 = 0.5625, \qquad P(RR\mid B) = 0.5^2 = 0.25. With equal priors the 0.5 cancels, so use the likelihoods directly: P(A\mid RR) = \frac{0…

Which urn? A Bayesian posterior from two draws

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