Question: Urn A1 contains 8 black and 2 white marbles. Urn A3 contains 3 black and 7 white marbles, and urn A3 contains 5 white and 5 black marbles. A fair die is to be cast. If the die turns up 1, 2 or 3 then a marble will be selected from A1. If the die turns up 4 or 5 a marble will be selected from A2. Finally, a marble will be selected from A3. If the die turns up 6. Given that the marble selected is black, what is the probability that the marble was from urn A2?

Solution:

Probability of marble being chosen from urn A1 = 3/6
Probability of selecting a black marble from A1 = 8/10
Hence, the joint probability of 1, 2 or 3 coming up in the fair die and then drawing a
black marble from A1 = 3/6 x 8/10 = 24/60.
Similarly, the probability of 4 or 5 turning up in the die and drawing a black marble from urn A2 = 2/6 x 3/10 = 6/60
and the probability of 6 turning up in the die and drawing a black marble from urn

A3 = 1/6 x 5/10 = 5/60

The probability of drawing a black marble from any of these urn is

24/60 + 6/60 + 5/60 = 35/60.

Assuming that the marble selected is black, the probability that the marble was chosen
from urn A2 is : P 6/60 / 35/60 = 6/35.