A bag A contains 3 white and 2 black balls and another bag B contains 2 white and 4 black balls. From a bag a ball is picked at random. The probability that the ball is black, is

Options

A. `2/7`
B. `7/9`
C. `4/15`
D. `8/15`

Answer

To find the probability of picking a black ball from either Bag A or Bag B, we can follow these steps:

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

### Step 2: Calculate the total number of balls in each bag
- Total balls in Bag A = 3 (white) + 2 (black) = 5 balls
- Total balls in Bag B = 2 (white) + 4 (black) = 6 balls

### Step 3: Determine the probability of selecting each bag
Since there are two bags, the probability of selecting either Bag A or Bag B is:
- Probability of selecting Bag A = 1/2
- Probability of selecting Bag B = 1/2

### Step 4: Calculate the probability of picking a black ball from Bag A
From Bag A, the probability of picking a black ball is:
- Number of black balls in Bag A = 2
- Total balls in Bag A = 5
- Probability of picking a black ball from Bag A = Number of black balls / Total balls = 2/5

### Step 5: Calculate the probability of picking a black ball from Bag B
From Bag B, the probability of picking a black ball is:
- Number of black balls in Bag B = 4
- Total balls in Bag B = 6
- Probability of picking a black ball from Bag B = Number of black balls / Total balls = 4/6 = 2/3

### Step 6: Use the law of total probability to find the overall probability of picking a black ball
The overall probability of picking a black ball can be calculated using:
\[ P(\text{Black}) = P(\text{A}) \cdot P(\text{Black | A}) + P(\text{B}) \cdot P(\text{Black | B}) \]

Substituting the values we have:
\[ P(\text{Black}) = \left(\frac{1}{2} \cdot \frac{2}{5}\right) + \left(\frac{1}{2} \cdot \frac{4}{6}\right) \]

### Step 7: Simplify the expression
Calculating each term:
1. From Bag A:
\[ \frac{1}{2} \cdot \frac{2}{5} = \frac{2}{10} = \frac{1}{5} \]
2. From Bag B:
\[ \frac{1}{2} \cdot \frac{4}{6} = \frac{4}{12} = \frac{1}{3} \]

Now, we need a common denominator to add these fractions:
- The common denominator of 5 and 3 is 15.

Converting the fractions:
- \( \frac{1}{5} = \frac{3}{15} \)
- \( \frac{1}{3} = \frac{5}{15} \)

Adding these:
\[ P(\text{Black}) = \frac{3}{15} + \frac{5}{15} = \frac{8}{15} \]

### Final Answer
The probability that the ball picked is black is \( \frac{8}{15} \).

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