There are two bags containing white and black balls. In the first bag, there are 8 white and 6 black balls and in the second bag, there are 4 white and 7 black balls. One ball is drawn at random from any of these two bags. Find the probability of this ball being black.

Options

A. `5/9`
B. `7/19`
C. `41/77`
D. `9/17`

Answer

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

### Step 1: Determine the total number of balls in each bag.
- **Bag 1**: 8 white balls + 6 black balls = 14 total balls.
- **Bag 2**: 4 white balls + 7 black balls = 11 total balls.

### Step 2: Calculate the probability of selecting each bag.
Since there are two bags, the probability of selecting either bag is:
- Probability of selecting Bag 1 = \( \frac{1}{2} \)
- Probability of selecting Bag 2 = \( \frac{1}{2} \)

### Step 3: Calculate the probability of drawing a black ball from Bag 1.
- The number of black balls in Bag 1 = 6.
- The total number of balls in Bag 1 = 14.
- Probability of drawing a black ball from Bag 1 = \( \frac{6}{14} = \frac{3}{7} \).

### Step 4: Calculate the probability of drawing a black ball from Bag 2.
- The number of black balls in Bag 2 = 7.
- The total number of balls in Bag 2 = 11.
- Probability of drawing a black ball from Bag 2 = \( \frac{7}{11} \).

### Step 5: Calculate the total probability of drawing a black ball.
Using the law of total probability:
\[ P(\text{Black}) = P(\text{Bag 1}) \times P(\text{Black from Bag 1}) + P(\text{Bag 2}) \times P(\text{Black from Bag 2}) \]
Substituting the values:
\[ P(\text{Black}) = \left(\frac{1}{2} \times \frac{3}{7}\right) + \left(\frac{1}{2} \times \frac{7}{11}\right) \]
Calculating each term:
\[ P(\text{Black}) = \frac{3}{14} + \frac{7}{22} \]

### Step 6: Find a common denominator to add the fractions.
The least common multiple of 14 and 22 is 154.
- Convert \( \frac{3}{14} \) to have a denominator of 154:
\[ \frac{3}{14} = \frac{3 \times 11}{14 \times 11} = \frac{33}{154} \]
- Convert \( \frac{7}{22} \) to have a denominator of 154:
\[ \frac{7}{22} = \frac{7 \times 7}{22 \times 7} = \frac{49}{154} \]

### Step 7: Add the two fractions.
\[ P(\text{Black}) = \frac{33}{154} + \frac{49}{154} = \frac{82}{154} \]

### Step 8: Simplify the fraction.
\[ \frac{82}{154} = \frac{41}{77} \quad (\text{dividing both numerator and denominator by 2}) \]

### Final Answer:
The probability of drawing a black ball is \( \frac{41}{77} \).

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