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 block balls otherwise a ball is chosen from urn B containing 3 white and 2 block balls then.
Q. The probability of drawing a block ball is

Options

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

Answer

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 is directly proportional to the number itself, we can express the probabilities as follows:

- Let \( P(1) = k \)
- \( P(2) = 2k \)
- \( P(3) = 3k \)
- \( P(4) = 4k \)
- \( P(5) = 5k \)
- \( P(6) = 6k \)

The sum of all probabilities must equal 1:

\[ P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1 \]

Substituting the expressions for each probability, we get:

\[ k + 2k + 3k + 4k + 5k + 6k = 1 \]

This simplifies to:

\[ 21k = 1 \implies k = \frac{1}{21} \]

### Step 2: Calculate the individual probabilities.

Now we can find the probabilities for each outcome:

- \( P(1) = \frac{1}{21} \)
- \( P(2) = \frac{2}{21} \)
- \( P(3) = \frac{3}{21} \)
- \( P(4) = \frac{4}{21} \)
- \( P(5) = \frac{5}{21} \)
- \( P(6) = \frac{6}{21} \)

### Step 3: Identify the prime numbers on the die.

The prime numbers from 1 to 6 are: 2, 3, and 5.

### Step 4: Calculate the probability of rolling a prime number.

The probability of rolling a prime number is:

\[ P(\text{prime}) = P(2) + P(3) + P(5) = \frac{2}{21} + \frac{3}{21} + \frac{5}{21} = \frac{10}{21} \]

### Step 5: Calculate the probability of rolling a non-prime number.

The probability of rolling a non-prime number is:

\[ P(\text{non-prime}) = 1 - P(\text{prime}) = 1 - \frac{10}{21} = \frac{11}{21} \]

### Step 6: Calculate the probability of drawing a black ball.

- If a prime number is rolled, a ball is chosen from urn A (which contains 2 white and 3 black balls). The probability of drawing a black ball from urn A is:

\[ P(\text{black | prime}) = \frac{3}{5} \]

- If a non-prime number is rolled, a ball is chosen from urn B (which contains 3 white and 2 black balls). The probability of drawing a black ball from urn B is:

\[ P(\text{black | non-prime}) = \frac{2}{5} \]

### Step 7: Calculate the total probability of drawing a black ball.

Using the law of total probability:

\[ P(\text{black}) = P(\text{prime}) \cdot P(\text{black | prime}) + P(\text{non-prime}) \cdot P(\text{black | non-prime}) \]

Substituting the values we have:

\[ P(\text{black}) = \left(\frac{10}{21} \cdot \frac{3}{5}\right) + \left(\frac{11}{21} \cdot \frac{2}{5}\right) \]

Calculating each term:

\[ P(\text{black}) = \frac{30}{105} + \frac{22}{105} = \frac{52}{105} \]

### Final Answer:

The probability of drawing a black ball is \( \frac{52}{105} \).

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