Assertion (A) : A fair die is thrown once. The probability of getting a prime number is $\frac{1}{2}$. Reason (R) : A natural number is a prime number if it has only two factors.

Step 1: Analyze Assertion (A)

First, let's analyze the assertion. When a fair die is thrown once, the possible outcomes are 1, 2, 3, 4, 5, and 6. We need to identify the prime numbers among these outcomes and then calculate the probability.

Step 2: Identify Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. From the possible outcomes {1, 2, 3, 4, 5, 6}, the prime numbers are 2, 3, and 5. The number 1 is not a prime number.

Step 3: Calculate Probability for Assertion (A)

There are 3 prime numbers (2, 3, 5) out of a total of 6 possible outcomes. Therefore, the probability of getting a prime number is $3/6$, which simplifies to $1/2$. This confirms that Assertion (A) is true.

Step 4: Analyze Reason (R)

Now, let's analyze the reason. The statement says, 'A natural number is a prime number if it has only two factors.' This is the correct definition of a prime number. For example, 2 has factors 1 and 2; 3 has factors 1 and 3; 5 has factors 1 and 5. The number 1 has only one factor (1), so it is not prime.

Step 5: Evaluate Relationship between A and R

Both Assertion (A) and Reason (R) are true statements. Furthermore, the definition of a prime number given in Reason (R) is directly used to identify the prime numbers in Assertion (A) to calculate the probability. Therefore, Reason (R) is the correct explanation for Assertion (A).

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