Which urn did the red ball come from?

Question

Two urns sit on a table. Urn A holds 3 red balls and 1 white ball. Urn B holds 1 red ball and 3 white balls. You pick an urn at random (each with probability $\tfrac12$ ), then draw one ball at random from it. The ball is red.

What is the probability that you drew from urn A?

Accepted Answer

Both urns hold 4 balls, so an easy way to see the answer is to imagine all the red balls as equally likely candidates for what you drew, weighted by how the draw actually happens. The chance of drawing red is 	frac34 from urn A and 	frac14 from urn B. With each urn chosen half the time, P(	ext{red}) = 	frac12\cdot	frac34 + 	frac12\cdot	frac14 = 	frac12. By Bayes' rule, P(A \mid 	ext{red}) = \frac{	frac12\cdot	frac34}{	frac12} = \frac34. Concretely: across the two urns there are 4 red balls total, 3 in A and 1 in B, and since the urns are equally likely and equally sized, each red …

Which urn did the red ball come from?

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