An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball drawn is black.

Answer

To find the probability that the third ball drawn from the urn is black, we can follow these steps:

### Step 1: Identify the initial conditions
The urn contains 2 white balls (W) and 2 black balls (B).

### Step 2: Determine the possible outcomes for drawing three balls
We can have the following cases for the colors of the balls drawn:
1. 2 Black and 1 White (BB W)
2. 2 White and 1 Black (WW B)

### Step 3: List the arrangements for each case
For each case, we can list the arrangements:
- For 2 Black and 1 White:
1. BBW
2. BWB
3. WBB

- For 2 White and 1 Black:
1. WWB
2. WBW
3. BWW

### Step 4: Calculate the probability for each arrangement
Now, we need to calculate the probability of each arrangement leading to the third ball being black.

#### Case 1: 2 Black and 1 White
- **Arrangement 1: BBW**
- P(B) = 2/4 (first ball black)
- P(B) = 2/4 (second ball black)
- P(W) = 2/4 (third ball white)
- Probability = (2/4) * (2/4) * (2/4) = 1/8

- **Arrangement 2: BWB**
- P(B) = 2/4 (first ball black)
- P(W) = 2/4 (second ball white)
- P(B) = 2/4 (third ball black)
- Probability = (2/4) * (2/4) * (2/4) = 1/8

- **Arrangement 3: WBB**
- P(W) = 2/4 (first ball white)
- P(B) = 2/4 (second ball black)
- P(B) = 2/4 (third ball black)
- Probability = (2/4) * (2/4) * (2/4) = 1/8

Total probability for Case 1 (2 Black, 1 White) = 1/8 + 1/8 + 1/8 = 3/8

#### Case 2: 2 White and 1 Black
- **Arrangement 1: WWB**
- P(W) = 2/4 (first ball white)
- P(W) = 1/3 (second ball white)
- P(B) = 2/2 (third ball black)
- Probability = (2/4) * (1/3) * (2/2) = 1/6

- **Arrangement 2: WBW**
- P(W) = 2/4 (first ball white)
- P(B) = 2/3 (second ball black)
- P(W) = 1/2 (third ball white)
- Probability = (2/4) * (2/3) * (1/2) = 1/6

- **Arrangement 3: BWW**
- P(B) = 2/4 (first ball black)
- P(W) = 2/3 (second ball white)
- P(W) = 1/2 (third ball white)
- Probability = (2/4) * (2/3) * (1/2) = 1/6

Total probability for Case 2 (2 White, 1 Black) = 1/6 + 1/6 + 1/6 = 1/2

### Step 5: Combine the probabilities
Now we combine the probabilities from both cases to find the total probability that the third ball drawn is black.

Total probability that the third ball is black = (3/8) + (1/2)
= (3/8) + (4/8) = 7/8

### Step 6: Final probability calculation
The probability that the third ball drawn is black is given by:

P(third ball is black) = (3/8) + (1/6) = (9/24) + (4/24) = 13/24