A bag contains 3 white balls and 2 black balls, another contains 5 white and 3 black balls, if a bag is chosen at random and a ball is drawn from it. What is the probability that it is white ?

Answer

To find the probability of drawing a white ball from one of the two bags, we can follow these steps:

### Step 1: Identify the contents of each bag
- **Bag A**: Contains 3 white balls and 2 black balls.
- **Bag B**: Contains 5 white balls and 3 black balls.

### Step 2: Calculate the total number of balls in each bag
- **Total in Bag A**: 3 (white) + 2 (black) = 5 balls
- **Total in Bag B**: 5 (white) + 3 (black) = 8 balls

### Step 3: Determine the probability of choosing each bag
Since a bag is chosen at random, the probability of choosing either bag is:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{2} \)

### Step 4: Calculate the probability of drawing a white ball from each bag
- **Probability of drawing a white ball from Bag A**:
\[ P(W|A) = \frac{\text{Number of white balls in A}}{\text{Total balls in A}} = \frac{3}{5} \]

- **Probability of drawing a white ball from Bag B**:
\[ P(W|B) = \frac{\text{Number of white balls in B}}{\text{Total balls in B}} = \frac{5}{8} \]

### Step 5: Use the law of total probability to find the overall probability of drawing a white ball
The overall probability of drawing a white ball can be calculated as follows:
\[ P(W) = P(A) \cdot P(W|A) + P(B) \cdot P(W|B) \]
Substituting the values we calculated:
\[ P(W) = \left(\frac{1}{2} \cdot \frac{3}{5}\right) + \left(\frac{1}{2} \cdot \frac{5}{8}\right) \]

### Step 6: Simplify the expression
Calculating each term:
- For Bag A:
\[ \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \]
- For Bag B:
\[ \frac{1}{2} \cdot \frac{5}{8} = \frac{5}{16} \]

Now, we need a common denominator to add these fractions. The least common multiple of 10 and 16 is 80.
- Convert \(\frac{3}{10}\) to a fraction with a denominator of 80:
\[ \frac{3}{10} = \frac{3 \times 8}{10 \times 8} = \frac{24}{80} \]
- Convert \(\frac{5}{16}\) to a fraction with a denominator of 80:
\[ \frac{5}{16} = \frac{5 \times 5}{16 \times 5} = \frac{25}{80} \]

Now we can add:
\[ P(W) = \frac{24}{80} + \frac{25}{80} = \frac{49}{80} \]

### Final Answer
The probability that the ball drawn is white is:
\[ \boxed{\frac{49}{80}} \]