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XAT 2014 — QA & DI Question 16

Simple EquationsEasy

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

Correct answer: 3 16 ​​​​​​​

  • A

    116    

  • B

    216    

  • 316​​​​​​​

  • D

    4 16

  • E

    None of the above

Solution

​​​​​​​

(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 (∵ (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 (∵ (X1) × (X2) = 25)

Y1 = 3 and Y2 = 5

∴ Y2 = 6

Probability of both marbles being blue = (5 × 3)/80

= 3/16

Hence, option (c).

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XAT 2014 QA & DI Q16: Aditya has a total of 18 red and blue marbles in two bags (each bag has marbles of both colors). A marble is r — Solution | TheCATExam