Three urns contain respectively 1 white, 2 black balls, 2W and 1B balls, 2 W and 2B balls. One ball is transferred from the first urn into the second, then one from the letter is transferred into the third. Finally one ball is drawn from the third urn. what is the probability of its being white ?

Answer

To solve the problem, we need to find the probability of drawing a white ball from the third urn after transferring balls between the urns. Let's break down the steps systematically.

### Step 1: Understand the contents of the urns
- **Urn 1**: 1 White (W), 2 Black (B) → Total = 3 balls
- **Urn 2**: 2 White (W), 1 Black (B) → Total = 3 balls
- **Urn 3**: 2 White (W), 2 Black (B) → Total = 4 balls

### Step 2: Transfer a ball from Urn 1 to Urn 2
There are two possible scenarios when transferring a ball from Urn 1 to Urn 2:
1. Transfer the White ball (W).
2. Transfer one of the Black balls (B).

#### Case 1: Transfer White Ball (W)
- Urn 1: 0W, 2B
- Urn 2: 3W, 1B → Total = 4 balls

#### Case 2: Transfer Black Ball (B)
- Urn 1: 1W, 1B
- Urn 2: 2W, 2B → Total = 4 balls

### Step 3: Transfer a ball from Urn 2 to Urn 3
Now we consider the outcomes of transferring a ball from Urn 2 to Urn 3 based on the previous cases.

#### From Case 1 (Transferred W):
- Probability of transferring W = 3/4 (3W out of 4 total)
- Urn 3: 2W, 2B → Total = 4 balls
- Probability of transferring B = 1/4 (1B out of 4 total)
- Urn 3: 3W, 2B → Total = 5 balls

#### From Case 2 (Transferred B):
- Probability of transferring W = 2/4 = 1/2 (2W out of 4 total)
- Urn 3: 2W, 2B → Total = 4 balls
- Probability of transferring B = 2/4 = 1/2 (2B out of 4 total)
- Urn 3: 1W, 3B → Total = 4 balls

### Step 4: Calculate the probability of drawing a white ball from Urn 3
Now we calculate the probability of drawing a white ball from Urn 3 for each scenario.

#### For Case 1:
1. **Transferred W from Urn 1 to Urn 2, then W from Urn 2 to Urn 3**:
- Probability = (3/4) * (2/4) = 3/8
- Probability of drawing W from Urn 3 = 2/4 = 1/2
- Contribution to final probability = (3/8) * (1/2) = 3/16

2. **Transferred W from Urn 1 to Urn 2, then B from Urn 2 to Urn 3**:
- Probability = (3/4) * (1/4) = 3/16
- Probability of drawing W from Urn 3 = 3/5
- Contribution to final probability = (3/16) * (3/5) = 9/80

#### For Case 2:
1. **Transferred B from Urn 1 to Urn 2, then W from Urn 2 to Urn 3**:
- Probability = (1/4) * (1/2) = 1/8
- Probability of drawing W from Urn 3 = 2/4 = 1/2
- Contribution to final probability = (1/8) * (1/2) = 1/16

2. **Transferred B from Urn 1 to Urn 2, then B from Urn 2 to Urn 3**:
- Probability = (1/4) * (1/2) = 1/8
- Probability of drawing W from Urn 3 = 1/4
- Contribution to final probability = (1/8) * (1/4) = 1/32

### Step 5: Sum all contributions to find the total probability
Now we add all contributions:
- From Case 1: 3/16 + 9/80
- From Case 2: 1/16 + 1/32

To add these fractions, we need a common denominator:
- The common denominator for 16, 80, and 32 is 80.

Calculating:
- 3/16 = 15/80
- 9/80 = 9/80
- 1/16 = 5/80
- 1/32 = 2.5/80 = 5/160 = 2.5/80

Now, summing:
- Total = (15 + 9 + 5 + 2.5) / 80 = 31.5 / 80 = 63 / 160

### Final Probability
The final probability of drawing a white ball from the third urn is:
\[ \frac{63}{160} \]