A bag 'A' contains 2 white and 3 red balls, a beg 'B' contains 4 white and 5 black balls. A bag is selected randomly and a ball is drawn from it. Drawn ball is observed to be white. Find the probability that bag 'B' was selected.

Answer

To solve the problem step by step, we will use Bayes' theorem to find the probability that bag B was selected given that a white ball was drawn.

### Step 1: Define the Events
Let:
- \( A \): Event that bag A is selected.
- \( B \): Event that bag B is selected.
- \( W \): Event that a white ball is drawn.

### Step 2: Calculate the Prior Probabilities
Since a bag is selected randomly:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{2} \)

### Step 3: Calculate the Probability of Drawing a White Ball from Each Bag
- From bag A, there are 2 white balls and 3 red balls, making a total of 5 balls. Thus, the probability of drawing a white ball from bag A is:
\[ P(W|A) = \frac{2}{5} \]

- From bag B, there are 4 white balls and 5 black balls, making a total of 9 balls. Thus, the probability of drawing a white ball from bag B is:
\[ P(W|B) = \frac{4}{9} \]

### Step 4: Calculate the Total Probability of Drawing a White Ball
Using the law of total probability:
\[ P(W) = P(W|A)P(A) + P(W|B)P(B) \]
Substituting the values:
\[ P(W) = \left(\frac{2}{5} \cdot \frac{1}{2}\right) + \left(\frac{4}{9} \cdot \frac{1}{2}\right) \]
Calculating each term:
\[ P(W) = \frac{2}{10} + \frac{4}{18} \]
Finding a common denominator (90):
\[ P(W) = \frac{18}{90} + \frac{20}{90} = \frac{38}{90} = \frac{19}{45} \]

### Step 5: Calculate the Probability that Bag B was Selected Given a White Ball was Drawn
Using Bayes' theorem:
\[ P(B|W) = \frac{P(W|B)P(B)}{P(W)} \]
Substituting the values:
\[ P(B|W) = \frac{\left(\frac{4}{9}\right) \left(\frac{1}{2}\right)}{\frac{19}{45}} \]
Calculating the numerator:
\[ P(B|W) = \frac{\frac{4}{18}}{\frac{19}{45}} = \frac{4 \cdot 45}{18 \cdot 19} \]
Simplifying:
\[ P(B|W) = \frac{180}{342} = \frac{10}{19} \]

### Final Answer
The probability that bag B was selected given that a white ball was drawn is:
\[ \boxed{\frac{10}{19}} \]