Three balls marked with 1, 2 and 3 are placed in an urn. One ball is drawn, its number is noted, then the ball is returned to the urn. This process is repeated and then repeated once more. Each ball is equally likely to be drawn on each occasion. If the sum of the number noted is 6, then the probability that the ball numbered with 2 is drawn at all the three occassion, is

Options

A. `(1)/(27)`
B. `(1)/(7)`
C. `(1)/(6)`
D. `(1)/(3)`

Explanation

To solve the problem, we need to determine the probability that the ball numbered with 2 is drawn on all three occasions, given that the sum of the numbers noted is 6.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have three balls marked with the numbers 1, 2, and 3. We draw a ball, note its number, and return it to the urn. This process is repeated three times. We need to find the probability that the ball numbered 2 is drawn all three times, given that the sum of the numbers drawn is 6.

2. **Finding Total Outcomes for Sum = 6:**
We need to find all the possible combinations of the numbers drawn that sum to 6. The possible combinations are:
- (1, 2, 3)
- (1, 3, 2)
- (2, 1, 3)
- (2, 3, 1)
- (3, 1, 2)
- (3, 2, 1)
- (2, 2, 2)

The first six combinations represent the different arrangements of drawing 1, 2, and 3, and the last combination represents drawing 2 three times.

Thus, the total number of ways to achieve a sum of 6 is 7.

3. **Finding Favorable Outcomes:**
The favorable outcome for our case is the scenario where the ball numbered 2 is drawn all three times, which is represented as (2, 2, 2). There is only one way to achieve this.

4. **Calculating the Probability:**
The probability is given by the formula:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]
Here, the number of favorable outcomes is 1 (the case (2, 2, 2)), and the total outcomes are 7. Therefore, the probability is:
\[ \text{Probability} = \frac{1}{7} \]

### Final Answer:
The probability that the ball numbered with 2 is drawn on all three occasions, given that the sum of the numbers noted is 6, is \(\frac{1}{7}\).