Financial Risk Manager Part 1

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%.