Financial Risk Manager Part 1
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The second bag has 3 red and 2 white balls. A bag is randomly selected, and a ball is drawn without replacement. 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?
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TTanishq
A
40%
B
50%
C
51%
D
53%
Explanation:
Explanation
This is a conditional probability problem that can be solved using the law of total probability:
Step 1: Calculate P(1st red)
- Bag 1: 5 red, 5 white → P(red|Bag1) = 5/10 = 0.5
- Bag 2: 3 red, 2 white → P(red|Bag2) = 3/5 = 0.6
- P(1st red) = (1/2) × (5/10) + (1/2) × (3/5) = 0.25 + 0.3 = 0.55
Step 2: Calculate P(2nd red | 1st red)
- If 1st red from Bag1: 4 red, 5 white remain → P(2nd red) = 4/9 ≈ 0.444
- If 1st red from Bag2: 2 red, 2 white remain → P(2nd red) = 2/4 = 0.5
- P(2nd red | 1st red) = (1/2) × (4/9) + (3/5) × (2/4) = 0.222 + 0.3 = 0.522
Step 3: Calculate P(2nd red | 1st white)
- If 1st white from Bag1: switch to Bag2 → P(2nd red) = 3/5 = 0.6
- If 1st white from Bag2: switch to Bag1 → P(2nd red) = 5/10 = 0.5
- P(2nd red | 1st white) = (1/2) × (3/5) + (2/5) × (5/10) = 0.3 + 0.2 = 0.5
Step 4: Calculate P(1st white)
- P(1st white) = 1 - P(1st red) = 1 - 0.55 = 0.45
Step 5: Apply law of total probability P(2nd red) = P(2nd red | 1st red) × P(1st red) + P(2nd red | 1st white) × P(1st white) = 0.522 × 0.55 + 0.5 × 0.45 = 0.2871 + 0.225 = 0.5121 ≈ 51%
Therefore, the probability that the second ball drawn is red is 51%.
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