Question

The probability of getting a prime number on rolling a fair die is:

• A. $\frac{1}{6}$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $\frac{2}{3}$

Hint:

Watch out for the number 1! It is a common trap to mistake 1 for a prime number. The smallest prime number is 2, which is also the only even prime number.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Probability measures the likelihood of an event occurring and is calculated as the ratio of favorable outcomes to the total number of possible outcomes. For a standard fair die, we need to list all possible outcomes and isolate which numbers are prime. A prime number is an integer greater than 1 whose only positive divisors are 1 and itself.

Step 2: Key Formula or Approach:
$$\text{Probability (P)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Step 3: Detailed Explanation:
When a fair six-sided die is rolled, the total possible outcomes form the sample space $S$: \[ S = \{1, 2, 3, 4, 5, 6\} \implies \text{Total possible outcomes} = 6 \] Identify the prime numbers within this sample space: 1 is neither prime nor composite. 2 is a prime number. 3 is a prime number. 4 is composite ($2 \times 2$). 5 is a prime number. 6 is composite ($2 \times 3$). The favorable outcomes (prime numbers) form the set $E = \{2, 3, 5\}$, which gives: \[ \text{Number of favorable outcomes} = 3 \] Calculate the probability: \[ P = \frac{3}{6} = \frac{1}{2} \]

Step 4: Final Answer:
The probability of getting a prime number is $\frac{1}{2}$.