Two bags contain 3 white, 2 black and 2 white, 4 black balls respectively. A ball is chosen at random then the probability of its being black is

Options

A. `8/(15)`
B. `2/3`
C. `6/4`
D. none

Answer

To solve the problem, we need to calculate the probability of drawing a black ball from two bags containing different numbers of black and white balls.

### Step-by-Step Solution:

1. **Identify the contents of the bags:**
- Bag 1 contains 3 white balls and 2 black balls.
- Bag 2 contains 2 white balls and 4 black balls.

2. **Calculate the total number of balls in each bag:**
- Total in Bag 1 = 3 (white) + 2 (black) = 5 balls.
- Total in Bag 2 = 2 (white) + 4 (black) = 6 balls.

3. **Determine the probability of selecting each bag:**
- Since there are 2 bags, the probability of selecting either bag is:
- P(A1) = Probability of selecting Bag 1 = 1/2
- P(A2) = Probability of selecting Bag 2 = 1/2

4. **Calculate the probability of drawing a black ball from each bag:**
- For Bag 1:
- P(B|A1) = Probability of drawing a black ball from Bag 1 = Number of black balls in Bag 1 / Total balls in Bag 1 = 2/5
- For Bag 2:
- P(B|A2) = Probability of drawing a black ball from Bag 2 = Number of black balls in Bag 2 / Total balls in Bag 2 = 4/6 = 2/3

5. **Use the law of total probability to find the overall probability of drawing a black ball:**
- P(B) = P(A1) * P(B|A1) + P(A2) * P(B|A2)
- P(B) = (1/2) * (2/5) + (1/2) * (2/3)

6. **Calculate each term:**
- First term: (1/2) * (2/5) = 2/10 = 1/5
- Second term: (1/2) * (2/3) = 2/6 = 1/3

7. **Find a common denominator to add the fractions:**
- The common denominator of 5 and 3 is 15.
- Convert the fractions:
- 1/5 = 3/15
- 1/3 = 5/15

8. **Add the two fractions:**
- P(B) = 3/15 + 5/15 = 8/15

9. **Conclusion:**
- The probability of drawing a black ball is 8/15.

### Final Answer:
The probability of drawing a black ball is **8/15**.