A person has 2 bags. He has 3 black and 4 white balls in one bag and 4 black and 3 white balls in another bag. Find the probability of getting a black ball.

Answer

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

### Step 1: Identify the contents of each bag.
- **Bag 1**: 3 black balls and 4 white balls.
- **Bag 2**: 4 black balls and 3 white balls.

### Step 2: Calculate the total number of balls in each bag.
- **Total in Bag 1**: 3 black + 4 white = 7 balls.
- **Total in Bag 2**: 4 black + 3 white = 7 balls.

### Step 3: Determine the probability of choosing each bag.
Since there are 2 bags, the probability of choosing either bag is:
- Probability of choosing Bag 1 = \( \frac{1}{2} \)
- Probability of choosing Bag 2 = \( \frac{1}{2} \)

### Step 4: Calculate the probability of drawing a black ball from each bag.
- **From Bag 1**: The probability of drawing a black ball = \( \frac{3 \text{ black balls}}{7 \text{ total balls}} = \frac{3}{7} \)
- **From Bag 2**: The probability of drawing a black ball = \( \frac{4 \text{ black balls}}{7 \text{ total balls}} = \frac{4}{7} \)

### Step 5: Use the law of total probability to find the overall probability of drawing a black ball.
The overall probability \( P(B) \) of drawing a black ball can be calculated as follows:
\[ P(B) = P(\text{Bag 1}) \times P(\text{Black ball from Bag 1}) + P(\text{Bag 2}) \times P(\text{Black ball from Bag 2}) \]
Substituting the values:
\[ P(B) = \left( \frac{1}{2} \times \frac{3}{7} \right) + \left( \frac{1}{2} \times \frac{4}{7} \right) \]
\[ P(B) = \frac{3}{14} + \frac{4}{14} = \frac{7}{14} = \frac{1}{2} \]

### Final Answer:
The probability of getting a black ball is \( \frac{1}{2} \).

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