There are two urns: one black, and one white, each of them with a certain number

Question

There are two urns: one black, and one white, each of them with a certain number of marbles. The black one contains 5 green marbles, and 3 red marbles. The white one contains 2 green marbles, and 6 red marbles. Let U be the event of selecting an urn, and assume: P(U = black) = 60%, and P(U = white) = 1 - P(U = black) = 40%. Now, let M be the outcome of selecting a marble. From the description above, we know that: P(M = green | U = black) = 5/8, P(M = red | U = black) = 3/8, P(M = green | U = white) = 2/8, P(M = red | U = white) = 6/8. Please answer the following questions: What is the probability of selecting a green marble? Given that in fact we selected a green marble, what is the probability that the green marble in fact came from the black urn?

Explanation / Answer

a) probabilty of selecting a green marble =P(G)=P(selecting black urn and choosing green marble+selecting white urn and selecting green marble) =P(U=black)*P(M=green|U=black)+P(U=white)*P(M=green|U=white)

=(0.6*(5/8)+0.4*(2/8)=0.475

b)probabilty we chose black urn, given green marble vcomes =P(U=black|M=green)=0.6*(5/8)/0.475=0.7895

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