There are two bags . One bag contains 3 white balls and 2 black balls . The other box contains 7 white and 3 balck balls , If a bag is selected at random and from it , then a ball is drawn then the probability that the ball is black is .

Answer

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

### Step 1: Understand the Problem
We have two bags:
- **Bag 1**: Contains 3 white balls and 2 black balls (Total = 5 balls)
- **Bag 2**: Contains 7 white balls and 3 black balls (Total = 10 balls)

### Step 2: Define Events
Let:
- \( E_1 \): Event of choosing Bag 1
- \( E_2 \): Event of choosing Bag 2
- \( A \): Event of drawing a black ball

### Step 3: Calculate the Probability of Choosing Each Bag
Since a bag is selected at random, the probabilities are:
- \( P(E_1) = \frac{1}{2} \)
- \( P(E_2) = \frac{1}{2} \)

### Step 4: Calculate the Probability of Drawing a Black Ball from Each Bag
- For **Bag 1**: The probability of drawing a black ball:
\[ P(A | E_1) = \frac{\text{Number of black balls in Bag 1}}{\text{Total number of balls in Bag 1}} = \frac{2}{5} \]

- For **Bag 2**: The probability of drawing a black ball:
\[ P(A | E_2) = \frac{\text{Number of black balls in Bag 2}}{\text{Total number of balls in Bag 2}} = \frac{3}{10} \]

### Step 5: Apply the Total Probability Theorem
The total probability of drawing a black ball \( P(A) \) can be calculated as:
\[ P(A) = P(E_1) \cdot P(A | E_1) + P(E_2) \cdot P(A | E_2) \]
Substituting the values:
\[ P(A) = \left(\frac{1}{2} \cdot \frac{2}{5}\right) + \left(\frac{1}{2} \cdot \frac{3}{10}\right) \]

### Step 6: Simplify the Expression
Calculating each term:
- First term:
\[ \frac{1}{2} \cdot \frac{2}{5} = \frac{2}{10} = \frac{1}{5} \]
- Second term:
\[ \frac{1}{2} \cdot \frac{3}{10} = \frac{3}{20} \]

Now, we need a common denominator to add these fractions:
- The common denominator of 5 and 20 is 20.
- Convert \( \frac{1}{5} \) to have a denominator of 20:
\[ \frac{1}{5} = \frac{4}{20} \]

Now, add the two fractions:
\[ P(A) = \frac{4}{20} + \frac{3}{20} = \frac{7}{20} \]

### Final Answer
The probability that the drawn ball is black is:
\[ \boxed{\frac{7}{20}} \]

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