A die is rolled and showing any number is directly proportional to that number. If prime number appears then a ball is chosen from urn `A` containing 2 white and 3 black balls otherwise a ball is chosen from urn `B` containing 3 white and 2 black balls Then.
The probability of drawing a black ball is

Options

A. `53/105`
B. `52/105`
C. `49/105`
D. `51/105`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the probabilities of rolling each number on the die.
Since the probability of rolling a number \( i \) is directly proportional to \( i \), we can express the probability as:
\[ P(i) = k \cdot i \]
where \( k \) is a constant.

### Step 2: Find the value of \( k \).
The total probability must sum to 1:
\[ P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = k(1 + 2 + 3 + 4 + 5 + 6) = k \cdot 21 = 1 \]
Thus,
\[ k = \frac{1}{21} \]

### Step 3: Calculate the probabilities for each outcome.
Using the value of \( k \):
- \( P(1) = \frac{1}{21} \)
- \( P(2) = \frac{2}{21} \)
- \( P(3) = \frac{3}{21} \)
- \( P(4) = \frac{4}{21} \)
- \( P(5) = \frac{5}{21} \)
- \( P(6) = \frac{6}{21} \)

### Step 4: Identify the prime numbers on the die.
The prime numbers between 1 and 6 are 2, 3, and 5.

### Step 5: Calculate the total probability of rolling a prime number.
\[ P(\text{prime}) = P(2) + P(3) + P(5) = \frac{2}{21} + \frac{3}{21} + \frac{5}{21} = \frac{10}{21} \]

### Step 6: Calculate the probability of rolling a non-prime number.
\[ P(\text{not prime}) = 1 - P(\text{prime}) = 1 - \frac{10}{21} = \frac{11}{21} \]

### Step 7: Determine the probabilities of drawing a black ball from each urn.
- **Urn A** (if a prime number is rolled): Contains 2 white and 3 black balls.
- Probability of drawing a black ball from Urn A:
\[ P(\text{black} | A) = \frac{3}{5} \]

- **Urn B** (if a non-prime number is rolled): Contains 3 white and 2 black balls.
- Probability of drawing a black ball from Urn B:
\[ P(\text{black} | B) = \frac{2}{5} \]

### Step 8: Use the law of total probability to find the overall probability of drawing a black ball.
\[ P(\text{black}) = P(A) \cdot P(\text{black} | A) + P(B) \cdot P(\text{black} | B) \]
Substituting the values:
\[ P(\text{black}) = P(\text{prime}) \cdot P(\text{black} | A) + P(\text{not prime}) \cdot P(\text{black} | B) \]
\[ = \left(\frac{10}{21} \cdot \frac{3}{5}\right) + \left(\frac{11}{21} \cdot \frac{2}{5}\right) \]
Calculating each term:
\[ = \frac{30}{105} + \frac{22}{105} = \frac{52}{105} \]

### Final Answer:
The probability of drawing a black ball is:
\[ \frac{52}{105} \]