A bag contains 3 black and 4 white balls. Second bag contains 5 black and 6 white balls. A ball is drawn at random from one of the bags and it is found to be black. What is the probability that this ball is drawn from second bag?

Explanation

To solve the problem, we will use Bayes' theorem to find the probability that the black ball was drawn from the second bag, given that a black ball was drawn.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( B_1 \) be the event that a ball is drawn from Bag 1.
- Let \( B_2 \) be the event that a ball is drawn from Bag 2.
- Let \( A \) be the event that a black ball is drawn.

2. **Calculate the probabilities of choosing each bag**:
Since both bags are equally likely to be chosen, we have:
\[ P(B_1) = P(B_2) = \frac{1}{2} \]

3. **Calculate the probability of drawing a black ball from each bag**:
- For Bag 1, which contains 3 black and 4 white balls (total 7 balls):
\[ P(A | B_1) = \frac{3}{7} \]
- For Bag 2, which contains 5 black and 6 white balls (total 11 balls):
\[ P(A | B_2) = \frac{5}{11} \]

4. **Calculate the total probability of drawing a black ball**:
Using the law of total probability:
\[ P(A) = P(A | B_1) \cdot P(B_1) + P(A | B_2) \cdot P(B_2) \]
Substituting the values:
\[ P(A) = \left(\frac{3}{7} \cdot \frac{1}{2}\right) + \left(\frac{5}{11} \cdot \frac{1}{2}\right) \]
\[ P(A) = \frac{3}{14} + \frac{5}{22} \]
To add these fractions, we need a common denominator, which is 77:
\[ P(A) = \frac{3 \cdot 11}{77} + \frac{5 \cdot 7}{77} = \frac{33 + 35}{77} = \frac{68}{77} \]

5. **Calculate the probability that the black ball was drawn from Bag 2**:
Using Bayes' theorem:
\[ P(B_2 | A) = \frac{P(A | B_2) \cdot P(B_2)}{P(A)} \]
Substituting the known values:
\[ P(B_2 | A) = \frac{\left(\frac{5}{11}\right) \cdot \left(\frac{1}{2}\right)}{\frac{68}{77}} \]
Simplifying:
\[ P(B_2 | A) = \frac{\frac{5}{22}}{\frac{68}{77}} = \frac{5}{22} \cdot \frac{77}{68} = \frac{5 \cdot 77}{22 \cdot 68} \]
Reducing:
\[ = \frac{385}{1496} \]
Simplifying further:
\[ = \frac{35}{136} \]

### Final Answer:
The probability that the black ball was drawn from the second bag is:
\[ \frac{35}{136} \]