Contents of the two urns is as given in this table. A fair die is tossed. If the face 1,2,4 or 5 comes, a marble is drawn from the urn A otherwise a marble is chosen from the urn B.
Urn Red Marbles White marbles Blue marbles
A 5 3 8
B 3 5 0
Let $E_{1} :$ Denote the event that a red marble is chosen
$E_{2}$ : Denote the event that a white marble is chosen
$E_{3}$ : Denote the event that a blue is chosen
Then

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The event $E_{1}, E_{2}$ and $E_{3}$ are equiprobable

If the marble drawn is red, the probability that it came from the urn A is $\frac{1}{2}$

If the marble drawn is white, the probability that the face 5 appeared on the die is $\frac{3}{32}$

$P (E_{1}), P (E_{2}), P (E_{3})$ are in A.P

answer is A, B, D.

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Detailed Solution

Urn	Red Marbles	White marbles	Blue marbles
A	5	3	8
B	3	5	0

$P (E_{1}) = P (R) = (\frac{2}{3}) (\frac{5}{16}) + (\frac{1}{3}) (\frac{3}{8}) = \frac{10}{48} + \frac{6}{48} = \frac{1}{3} P (E_{2}) = P (W) = (\frac{2}{3}) (\frac{3}{16}) + (\frac{1}{3}) (\frac{5}{8}) = \frac{6}{48} + \frac{10}{48} = \frac{1}{3} P (E_{3}) = P (B) = (\frac{2}{3}) (\frac{8}{16}) = \frac{1}{3}$