Financial Risk Manager Part 1
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There are two bags with red and white balls. The first bag has 5 red and 5 white balls. The second bag has 3 red and 2 white balls. A bag is randomly selected and a ball is drawn. If the ball is red, another ball is selected from the same bag. If the ball is white, another ball is selected from the other bag. What is the probability that the second ball drawn is red?
Other
Community
NNikitesh
A
40%
B
50%
C
51%
D
53%
Explanation:
The correct answer is C (51%).
Explanation:
We need to calculate P(2nd ball is red) using conditional probability and the law of total probability.
Step 1: Calculate P(1st ball is red)
- Probability of selecting Bag 1 = 1/2
- Probability of drawing red from Bag 1 = 5/10 = 1/2
- Probability of selecting Bag 2 = 1/2
- Probability of drawing red from Bag 2 = 3/5
P(1st red) = (1/2 × 1/2) + (1/2 × 3/5) = 1/4 + 3/10 = 0.25 + 0.30 = 0.55
Step 2: Calculate P(1st ball is white) P(1st white) = 1 - P(1st red) = 1 - 0.55 = 0.45
Step 3: Calculate P(2nd red | 1st red) If the first ball is red, we draw the second ball from the same bag.
Case 1: First ball red from Bag 1
- Probability: (1/2 × 1/2) / 0.55 = 0.25/0.55 = 5/11
- After drawing one red from Bag 1: 4 red, 5 white remain
- P(2nd red | Bag 1, 1st red) = 4/9
Case 2: First ball red from Bag 2
- Probability: (1/2 × 3/5) / 0.55 = 0.30/0.55 = 6/11
- After drawing one red from Bag 2: 2 red, 2 white remain
- P(2nd red | Bag 2, 1st red) = 2/4 = 1/2
P(2nd red | 1st red) = (5/11 × 4/9) + (6/11 × 1/2) = (20/99) + (6/22) ≈ 0.2020 + 0.2727 = 0.4747
Step 4: Calculate P(2nd red | 1st white) If the first ball is white, we draw the second ball from the other bag.
Case 1: First ball white from Bag 1
- Probability: (1/2 × 1/2) / 0.45 = 0.25/0.45 = 5/9
- Then draw from Bag 2 (original composition: 3 red, 2 white)
- P(2nd red | Bag 2) = 3/5 = 0.6
Case 2: First ball white from Bag 2
- Probability: (1/2 × 2/5) / 0.45 = 0.20/0.45 = 4/9
- Then draw from Bag 1 (original composition: 5 red, 5 white)
- P(2nd red | Bag 1) = 5/10 = 0.5
P(2nd red | 1st white) = (5/9 × 3/5) + (4/9 × 1/2) = (15/45) + (4/18) = 1/3 + 2/9 = 3/9 + 2/9 = 5/9 ≈ 0.5556
Step 5: Calculate total probability using law of total probability P(2nd red) = P(2nd red | 1st red) × P(1st red) + P(2nd red | 1st white) × P(1st white) = 0.4747 × 0.55 + 0.5556 × 0.45 = 0.2611 + 0.2500 = 0.5111 ≈ 51.11%
This matches option C (51%).
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