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

Answer

To solve the problem, we need to find the probability of drawing a red ball (event A) given that a blue ball (event B) has already been drawn from the urn.

### Step-by-Step Solution:

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

**Hint**: Always start by determining the total number of items in the set.

2. **Understand the Scenario After Drawing a Blue Ball**:
- Since we are drawing without replacement, after drawing one blue ball, the number of balls left in the urn will be:
- Total balls left = 15 - 1 = 14 balls.
- The number of red balls remains the same, which is 6 red balls.

**Hint**: Remember that drawing without replacement changes the total count of items but not the count of the remaining items of the same type.

3. **Calculate the Probability of Drawing a Red Ball After Drawing a Blue Ball**:
- We need to find the probability of event A (drawing a red ball) given event B (a blue ball has been drawn).
- The probability can be expressed as:
\[ P(A|B) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes after B}} \]
- Here, the number of favorable outcomes for A (drawing a red ball) = 6 (since all red balls are still in the urn).
- The total outcomes after drawing a blue ball = 14 (total balls left).

**Hint**: Use the conditional probability formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \) when applicable, but in this case, you can directly calculate since we know the outcomes.

4. **Substituting the Values**:
- Now substituting the values into the formula:
\[ P(A|B) = \frac{6}{14} \]

5. **Simplifying the Fraction**:
- Simplifying \( \frac{6}{14} \):
\[ P(A|B) = \frac{3}{7} \]

**Hint**: Always simplify your fractions to their lowest terms for clarity.

### Final Answer:
The probability of drawing a red ball after a blue ball has been drawn is \( \frac{3}{7} \).