Question: A bag contains 3 red and 2 black ball. Another bag contains 4 red and 5 black balls. A ball is drawn from the first bag and is placed in the second. A ball is then drawn from the second. What is the probability that this draw of red ball is due to the drawing of red ball from the first bag ?

1. 3/5
2. 2/5
3. 4/25
4. 3/10
5. 15/23

Correct Answer: E
Solution and Explanation:

Approach Solution 1:
Given:
Bag 1: 3 Red, 2 Black
Bag 2: 4 Red, 5 Black

1st transaction:
A ball is moved from bag 1 and shifted to bag 2
P(R)= 3/5 and P(B)= ⅖

2nd Transaction:
A ball is picked from Bag 2.

Required: Probability that this draw of red ball is due to the drawing of red ball from the first bag

Case 1: Black ball was picked from Bag 1 and then a Red is picked from Bag 2
P(R) = 2/5∗4/10= 8/50
Case 2: Red ball was picked from Bag 1 and then a Red is picked from Bag 2
P(R)= 3/5*5/10= 15/50

We need the probability of occurrence off Case 2:
Hence P(Case 2)= 15/50 / (8/50+15/50)= 15/23

Approach Solution 2:
Probability of getting red ball from bag1 and then red from bag 2 = 3/5*5/10 = 15/50
Probability of getting black ball from bag1 and then red from bag 2 = 2/5*4/10 = 8/50

Now comes the condition, hence apply conditional probability:
P(Red)/ P(Red + Black)
(15/50)/(15/50+8/50) = 15/50 /23/50 = 15/23.

Approach Solution 3:
The ball transferred from the first bag to the second could be a black ball or a red ball. Therefore, we have two scenarios:

1) A red ball is drawn from the 2nd bag after a black ball is drawn from 1st bag and transferred to the 2nd bag.
2) A red ball is drawn from the 2nd bag after a red ball is drawn from 1st bag and transferred to the 2nd bag.

Let’s find the probability of each scenario:

1) P(1st = black, 2nd = red) = 2/5 x 4/10 = 8/50
2) P(1st = red, 2nd = red) = 3/5 x 5/10 = 15/50

Thus the probability of drawing a red ball from the 2nd bag is 8/50 + 15/50 = 23/50, and the probability that it is due to a red ball having been drawn from the 1st bag is:
(15/50)/(23/50) = 15/23

“A bag contains 3 red and 2 black ball. Another bag contains 4 red and”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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