Two urns A and B contain 6 black and 4 white, 4 black and 6 white balls respectively. Two balls are drawn from one of the urns. If both the balls drawn are white, find the probability that the balls are drawn from urn B.

Answer

To solve the problem, we will use Bayes' theorem. We need to find the probability that the balls drawn are from urn B given that both balls drawn are white.

### Step 1: Define Events
Let:
- \( E_1 \): Event that urn A is chosen.
- \( E_2 \): Event that urn B is chosen.
- \( W \): Event that both balls drawn are white.

### Step 2: Calculate Prior Probabilities
Since we have two urns and we choose one at random:
\[ P(E_1) = P(E_2) = \frac{1}{2} \]

### Step 3: Calculate Probability of Drawing Two White Balls from Each Urn
1. **For urn A** (which has 6 black and 4 white balls):
- Total balls = 10
- Probability of drawing 2 white balls:
\[ P(W | E_1) = \frac{\text{Number of ways to choose 2 white from 4}}{\text{Total ways to choose 2 from 10}} = \frac{\binom{4}{2}}{\binom{10}{2}} = \frac{6}{45} = \frac{2}{15} \]

2. **For urn B** (which has 4 black and 6 white balls):
- Total balls = 10
- Probability of drawing 2 white balls:
\[ P(W | E_2) = \frac{\text{Number of ways to choose 2 white from 6}}{\text{Total ways to choose 2 from 10}} = \frac{\binom{6}{2}}{\binom{10}{2}} = \frac{15}{45} = \frac{1}{3} \]

### Step 4: Calculate Total Probability of Drawing Two White Balls
Using the law of total probability:
\[ P(W) = P(W | E_1)P(E_1) + P(W | E_2)P(E_2) \]
Substituting the values:
\[ P(W) = \left(\frac{2}{15} \cdot \frac{1}{2}\right) + \left(\frac{1}{3} \cdot \frac{1}{2}\right) \]
Calculating each term:
\[ = \frac{2}{30} + \frac{5}{30} = \frac{7}{30} \]

### Step 5: Apply Bayes' Theorem
We want to find \( P(E_2 | W) \):
\[ P(E_2 | W) = \frac{P(W | E_2)P(E_2)}{P(W)} \]
Substituting the values:
\[ P(E_2 | W) = \frac{\left(\frac{1}{3}\right) \left(\frac{1}{2}\right)}{\frac{7}{30}} = \frac{\frac{1}{6}}{\frac{7}{30}} = \frac{30}{42} = \frac{5}{7} \]

### Final Answer
The probability that the balls drawn are from urn B given that both balls drawn are white is:
\[ \boxed{\frac{5}{7}} \]