A white ball, which urn did it come from?

Question

Two urns sit on a table: - Urn 1: 3 white balls, 1 black ball. - Urn 2: 1 white ball, 3 black balls. You pick an urn at random (50/50), reach in without looking, and draw a white ball. What is the probability you drew from Urn 1?

Accepted Answer

Let the hypotheses be Urn 1 and Urn 2, each with prior 	frac12. The chance of drawing white from each: P(	ext{white}\mid 	ext{Urn 1}) = 	frac34,\qquad P(	ext{white}\mid 	ext{Urn 2}) = 	frac14. Bayes numerators: 	ext{Urn 1: } 	frac12	imes	frac34 = 	frac38,\qquad 	ext{Urn 2: } 	frac12	imes	frac14 = 	frac18. Normalize by their sum 	frac38 + 	frac18 = 	frac48: P(	ext{Urn 1}\mid 	ext{white}) = \frac{3/8}{4/8} = \frac{3}{4}. Since the urns were equally likely a priori, the white ball is 3-to-1 more suggestive of the white-heavy urn. Takeaway. With equal priors, the posterior i…

A white ball, which urn did it come from?

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