Two bags, a red marble, and reverse detective work

Question

Bag A holds 3 red and 2 white marbles. Bag B holds 1 red and 4 white marbles. You flip a fair coin to decide which bag to reach into, then draw one marble at random. It comes out red.

What is the probability the marble came from Bag A?

Accepted Answer

The coin makes each bag equally likely to start with, so imagine running the experiment 100 times: about 50 draws from Bag A and 50 from Bag B. - From Bag A: the chance of red is 3/5, so about 50 	imes 3/5 = 30 red marbles. - From Bag B: the chance of red is 1/5, so about 50 	imes 1/5 = 10 red marbles. Out of the 30 + 10 = 40 times you pull a red marble, 30 of them came from Bag A: P(	ext{Bag A} \mid 	ext{red}) = \frac{30}{40} = \frac{3}{4}. A red draw is stronger evidence for Bag A because Bag A is much richer in red marbles, so seeing red shifts your belief from an even 50-50 up to 75 p…

Two bags, a red marble, and reverse detective work

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