One bag contains 5 white balls and 3 black balls and a second bag contains 2 white balls and 4 black balls. One ball drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from second bag is black?

Explanation

To solve the problem step by step, we need to calculate the probability of drawing a black ball from the second bag after transferring one ball from the first bag to the second bag.

### Step 1: Identify the contents of both bags.
- **Bag 1**: 5 white balls and 3 black balls (Total = 8 balls)
- **Bag 2**: 2 white balls and 4 black balls (Total = 6 balls)

### Step 2: Determine the probabilities of transferring a ball from Bag 1 to Bag 2.
1. **Probability of transferring a white ball from Bag 1**:
- There are 5 white balls and 3 black balls in Bag 1.
- Probability of transferring a white ball (P(W)) = Number of white balls / Total balls in Bag 1 = 5/8.

2. **Probability of transferring a black ball from Bag 1**:
- Probability of transferring a black ball (P(B)) = Number of black balls / Total balls in Bag 1 = 3/8.

### Step 3: Analyze the two cases based on the transferred ball.

#### Case 1: A white ball is transferred to Bag 2.
- After transferring a white ball, Bag 2 will have:
- White balls = 2 + 1 = 3
- Black balls = 4
- Total balls in Bag 2 = 3 + 4 = 7

- **Probability of drawing a black ball from Bag 2**:
- P(Black | White transferred) = Number of black balls in Bag 2 / Total balls in Bag 2 = 4/7.

- **Total probability for Case 1**:
- P(Case 1) = P(W) * P(Black | White transferred) = (5/8) * (4/7) = 20/56.

#### Case 2: A black ball is transferred to Bag 2.
- After transferring a black ball, Bag 2 will have:
- White balls = 2
- Black balls = 4 + 1 = 5
- Total balls in Bag 2 = 2 + 5 = 7

- **Probability of drawing a black ball from Bag 2**:
- P(Black | Black transferred) = Number of black balls in Bag 2 / Total balls in Bag 2 = 5/7.

- **Total probability for Case 2**:
- P(Case 2) = P(B) * P(Black | Black transferred) = (3/8) * (5/7) = 15/56.

### Step 4: Calculate the total probability of drawing a black ball from Bag 2.
- Total probability = P(Case 1) + P(Case 2) = (20/56) + (15/56) = (20 + 15) / 56 = 35/56.

### Final Answer:
The probability that a ball drawn from the second bag is black is **35/56**.

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