XAT 2014 — QA & DI Question 16
Aditya has a total of 18 red and blue marbles in two bags (each bag has marbles of both colors). A marble is randomly drawn from the first bag followed by another randomly drawn from the second bag, the probability of both being red is 5/16. What is the probability of both marbles being blue?
Answer & solution
- A
- B
âââââââ
- D
4 16
- E
None of the above
âââââââ
(X1 + Y1) + (X2 + Y2) = 18 … (i)
Selecting a marble from the first bag and then from the second bag can be done in (X1 + Y1) × (X2 + Y2) ways.
Selecting a red marble from the first bag and then a red marble from the second bag can be done in (X1) × (X2) ways.
∴ Probability of selecting red marbles from both the bags = (X1) × (X2)/ (X1 + Y1) × (X2 + Y2) = 5/16
Let (X1 + Y1) × (X2 + Y2) = 16a … (ii)
∴ (X1) × (X2) = 5a … (iii)
Considering (i) and (ii), a = 2 or 5
Case I: a = 2
(X1 + Y1) × (X2 + Y2) = 32 … (iv)
∴ From (i) and (iv),
(X1 + Y1) = 2 and (X2 + Y2) = 16
∴ X1 = Y1 = 1 ⇒ X2 = 10 (âµ (X1) × (X2) = 10)
∴ Y2 = 6
Probability of both marbles being blue = (1 × 6)/32
= 3/16
Case II: a = 5
(X1 + Y1) × (X2 + Y2) = 80 … (v)
∴ From (i) and (v),
(X1 + Y1) = 8 and (X2 + Y2) = 10
∴ X1 = X2 = 5 (âµ (X1) × (X2) = 25)
Y1 = 3 and Y2 = 5
∴ Y2 = 6
Probability of both marbles being blue = (5 × 3)/80
= 3/16
Hence, option (c).