There are 3 red and 2 black, 2 red and 3 black, 4 red and 1 black balls in three bags respectively. The probability of selecting each bag is same. A ball is drawn at random from one bag. Find the probability that it is black.

Explanation

To solve the problem step by step, we will calculate the probability of drawing a black ball from each of the three bags and then combine these probabilities to find the overall probability of drawing a black ball.

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

### Step 2: Determine the probability of selecting each bag
Since the probability of selecting each bag is the same, we have:
- Probability of selecting Bag 1 = \( \frac{1}{3} \)
- Probability of selecting Bag 2 = \( \frac{1}{3} \)
- Probability of selecting Bag 3 = \( \frac{1}{3} \)

### Step 3: Calculate the probability of drawing a black ball from each bag
1. **For Bag 1**:
- Total balls = 3 red + 2 black = 5
- Probability of drawing a black ball = \( \frac{2}{5} \)

2. **For Bag 2**:
- Total balls = 2 red + 3 black = 5
- Probability of drawing a black ball = \( \frac{3}{5} \)

3. **For Bag 3**:
- Total balls = 4 red + 1 black = 5
- Probability of drawing a black ball = \( \frac{1}{5} \)

### Step 4: Combine the probabilities using the law of total probability
The overall probability of drawing a black ball can be calculated as follows:
\[ P(\text{Black}) = P(\text{Bag 1}) \cdot P(\text{Black | Bag 1}) + P(\text{Bag 2}) \cdot P(\text{Black | Bag 2}) + P(\text{Bag 3}) \cdot P(\text{Black | Bag 3}) \]

Substituting the values:
\[ P(\text{Black}) = \left(\frac{1}{3} \cdot \frac{2}{5}\right) + \left(\frac{1}{3} \cdot \frac{3}{5}\right) + \left(\frac{1}{3} \cdot \frac{1}{5}\right) \]

### Step 5: Simplify the expression
Calculating each term:
- From Bag 1: \( \frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15} \)
- From Bag 2: \( \frac{1}{3} \cdot \frac{3}{5} = \frac{3}{15} \)
- From Bag 3: \( \frac{1}{3} \cdot \frac{1}{5} = \frac{1}{15} \)

Now, combine these probabilities:
\[ P(\text{Black}) = \frac{2}{15} + \frac{3}{15} + \frac{1}{15} = \frac{6}{15} \]

### Step 6: Simplify the final probability
\[ P(\text{Black}) = \frac{6}{15} = \frac{2}{5} \]

### Final Answer
The probability that the ball drawn is black is \( \frac{2}{5} \).