An urn contains 6 red balls and 9 green balls . Two balls are drawn from the urn one after other without replacement .Find the probability of drawing a red ball when a green ball has been drawn from the urn .

Options

A. `(3)/(15)`
B. `(3)/(7)`
C. `(9)/(14)`
D. `(7)/(11)`

Answer

To find the probability of drawing a red ball after a green ball has been drawn from the urn, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls**:
- There are 6 red balls and 9 green balls in the urn.
- Total number of balls = 6 (red) + 9 (green) = 15 balls.

2. **Understand the Drawing Process**:
- We are drawing two balls one after the other without replacement.
- We want to find the probability of drawing a red ball (event A) after a green ball has been drawn first (event B).

3. **Calculate the Probability of Drawing a Green Ball First**:
- The probability of drawing a green ball first (event B) is:
\[ P(B) = \frac{\text{Number of green balls}}{\text{Total number of balls}} = \frac{9}{15} = \frac{3}{5} \]

4. **Calculate the Remaining Balls After Drawing a Green Ball**:
- After drawing one green ball, the total number of balls left in the urn is:
\[ 15 - 1 = 14 \text{ balls} \]
- The number of red balls remains the same, which is 6.

5. **Calculate the Probability of Drawing a Red Ball After a Green Ball**:
- The probability of drawing a red ball after a green ball has been drawn is:
\[ P(A | B) = \frac{\text{Number of red balls}}{\text{Total number of remaining balls}} = \frac{6}{14} = \frac{3}{7} \]

6. **Final Result**:
- Therefore, the probability of drawing a red ball after a green ball has been drawn is:
\[ P(A | B) = \frac{3}{7} \]

### Summary:
The probability of drawing a red ball after a green ball has been drawn from the urn is \( \frac{3}{7} \).