A box X contains 1 white ball, 3 red balls and 2 black balls. Another box Y contains 2 white balls, 3 red balls and 4 black balls. If one ball is drawn from each of the two boxes, then the probability the balls are of different colours is

Options

A. `19/54`
B. `35/54`
C. `17/54`
D. `37/54`

Answer

To find the probability that the balls drawn from the two boxes are of different colors, we can follow these steps:

### Step 1: Identify the total number of balls in each box.
- Box X contains:
- 1 White ball
- 3 Red balls
- 2 Black balls
- **Total in Box X = 1 + 3 + 2 = 6 balls**

- Box Y contains:
- 2 White balls
- 3 Red balls
- 4 Black balls
- **Total in Box Y = 2 + 3 + 4 = 9 balls**

### Step 2: Calculate the total number of outcomes when drawing one ball from each box.
- The total number of outcomes when drawing one ball from Box X and one from Box Y is:
\[ \text{Total Outcomes} = \text{Total in Box X} \times \text{Total in Box Y} = 6 \times 9 = 54 \]

### Step 3: Calculate the number of favorable outcomes for different colors.
We will consider the different cases based on the color of the ball drawn from Box X and the color of the ball drawn from Box Y.

1. **Case 1: White from Box X and Red or Black from Box Y**
- Probability of drawing White from Box X = \( \frac{1}{6} \)
- Probability of drawing Red or Black from Box Y = \( \frac{3 + 4}{9} = \frac{7}{9} \)
- Favorable outcomes for this case = \( 1 \times 7 = 7 \)

2. **Case 2: Red from Box X and White or Black from Box Y**
- Probability of drawing Red from Box X = \( \frac{3}{6} = \frac{1}{2} \)
- Probability of drawing White or Black from Box Y = \( \frac{2 + 4}{9} = \frac{6}{9} = \frac{2}{3} \)
- Favorable outcomes for this case = \( 3 \times 6 = 18 \)

3. **Case 3: Black from Box X and White or Red from Box Y**
- Probability of drawing Black from Box X = \( \frac{2}{6} = \frac{1}{3} \)
- Probability of drawing White or Red from Box Y = \( \frac{2 + 3}{9} = \frac{5}{9} \)
- Favorable outcomes for this case = \( 2 \times 5 = 10 \)

### Step 4: Calculate the total number of favorable outcomes.
- Total favorable outcomes = \( 7 + 18 + 10 = 35 \)

### Step 5: Calculate the probability of drawing balls of different colors.
- The probability \( P \) that the balls are of different colors is given by:
\[ P(\text{Different Colors}) = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{35}{54} \]

Thus, the probability that the balls drawn from the two boxes are of different colors is \( \frac{35}{54} \).

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