A bag A contains 3 white and 2 black balls and another bag B contains 2 white and 4 black balls. A bag and a ball out of it are picked at random. What is the probability that the ball is white?

Answer

To solve the problem step by step, we will calculate the probability of selecting a white ball from either bag A or bag B.

### Step 1: Identify the contents of the bags
- Bag A contains 3 white balls and 2 black balls.
- Bag B contains 2 white balls and 4 black balls.

### Step 2: Calculate the total number of balls in each bag
- Total balls in Bag A = 3 (white) + 2 (black) = 5 balls
- Total balls in Bag B = 2 (white) + 4 (black) = 6 balls

### Step 3: Calculate the probability of selecting each bag
Since there are two bags (A and B), the probability of selecting either bag is:
- Probability of selecting Bag A = 1/2
- Probability of selecting Bag B = 1/2

### Step 4: Calculate the probability of selecting a white ball from each bag
- Probability of selecting a white ball from Bag A = Number of white balls in Bag A / Total balls in Bag A = 3/5
- Probability of selecting a white ball from Bag B = Number of white balls in Bag B / Total balls in Bag B = 2/6 = 1/3

### Step 5: Use the law of total probability to find the overall probability of selecting a white ball
The overall probability of selecting a white ball can be calculated as follows:
\[ P(\text{White}) = P(\text{Bag A}) \times P(\text{White | Bag A}) + P(\text{Bag B}) \times P(\text{White | Bag B}) \]
Substituting the values we have:
\[ P(\text{White}) = \left(\frac{1}{2} \times \frac{3}{5}\right) + \left(\frac{1}{2} \times \frac{1}{3}\right) \]

### Step 6: Simplify the expression
Calculating each term:
- From Bag A: \(\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}\)
- From Bag B: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\)

Now, we need a common denominator to add these two fractions. The least common multiple of 10 and 6 is 30.

Converting both fractions:
- \(\frac{3}{10} = \frac{9}{30}\)
- \(\frac{1}{6} = \frac{5}{30}\)

Adding these:
\[ P(\text{White}) = \frac{9}{30} + \frac{5}{30} = \frac{14}{30} = \frac{7}{15} \]

### Final Answer
The probability that the ball picked is white is \(\frac{7}{15}\).

---