A bag contains 4 white and 2 black balls .Another contains 3 white and 5 black balls .One ball is drawn from each bag .
Find the probability that one is white and one is black .

Options

A. `(11)/(24)`
B. `(13)/(24)`
C. `(1)/(2)`
D. `(1)/(6)`

Answer

To solve the problem of finding the probability that one ball drawn from each bag is white and the other is black, we can break it down into clear steps.

### Step 1: Identify the total number of balls in each bag.
- **Bag 1** contains 4 white and 2 black balls.
- Total in Bag 1 = 4 + 2 = 6 balls.
- **Bag 2** contains 3 white and 5 black balls.
- Total in Bag 2 = 3 + 5 = 8 balls.

### Step 2: Calculate the probability of drawing a white ball from Bag 1 and a black ball from Bag 2.
- Probability of drawing a white ball from Bag 1:
\[ P(\text{White from Bag 1}) = \frac{4}{6} = \frac{2}{3} \]
- Probability of drawing a black ball from Bag 2:
\[ P(\text{Black from Bag 2}) = \frac{5}{8} \]
- Combined probability for this scenario:
\[ P(\text{White from Bag 1 and Black from Bag 2}) = P(\text{White from Bag 1}) \times P(\text{Black from Bag 2}) = \frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12} \]

### Step 3: Calculate the probability of drawing a black ball from Bag 1 and a white ball from Bag 2.
- Probability of drawing a black ball from Bag 1:
\[ P(\text{Black from Bag 1}) = \frac{2}{6} = \frac{1}{3} \]
- Probability of drawing a white ball from Bag 2:
\[ P(\text{White from Bag 2}) = \frac{3}{8} \]
- Combined probability for this scenario:
\[ P(\text{Black from Bag 1 and White from Bag 2}) = P(\text{Black from Bag 1}) \times P(\text{White from Bag 2}) = \frac{1}{3} \times \frac{3}{8} = \frac{3}{24} = \frac{1}{8} \]

### Step 4: Add the probabilities from Step 2 and Step 3.
- Total probability of getting one white and one black ball:
\[ P(\text{One white and one black}) = P(\text{White from Bag 1 and Black from Bag 2}) + P(\text{Black from Bag 1 and White from Bag 2} \]
\[ = \frac{5}{12} + \frac{1}{8} \]
- To add these fractions, find a common denominator (which is 24):
\[ = \frac{5 \times 2}{12 \times 2} + \frac{1 \times 3}{8 \times 3} = \frac{10}{24} + \frac{3}{24} = \frac{13}{24} \]

### Final Answer:
The probability that one ball is white and one ball is black is:
\[ \frac{13}{24} \]