Step 1: Understanding the Question:
The topic for this question is Probability. Specifically, it involves a "Single Event" experiment: rolling a six-sided fair die. Probability represents the likelihood of a specific event occurring compared to all possible outcomes. In this case, the target event is landing on a "prime number." To solve this, we need to correctly identify which integers on a die are prime.
Step 2 : Key Formulas and approach:
1. Probability Formula: $P(E) = \frac{n(E)}{n(S)}$, where $n(E)$ is the number of favorable outcomes and $n(S)$ is the total number of possible outcomes.
2. Prime Number Definition: A number greater than 1 that has exactly two divisors: 1 and itself.
The approach involves listing the sample space, filtering for primes, and simplifying the resulting fraction.
Step 3 : Detailed Explanation:
A standard fair die has six faces numbered 1, 2, 3, 4, 5, and 6. Therefore, our total sample space $S = \{1, 2, 3, 4, 5, 6\}$. This gives us $n(S) = 6$.
Next, we examine each number to see if it is a prime number.
The number 1 is a special case; it is neither prime nor composite because it only has one divisor. Many students mistakenly include 1 as a prime.
The number 2 is prime (divisible by 1 and 2). In fact, it is the only even prime number.
The number 3 is prime (divisible by 1 and 3).
The number 4 is composite ($2 \times 2 = 4$).
The number 5 is prime (divisible by 1 and 5).
The number 6 is composite ($2 \times 3 = 6$).
Based on this check, our set of favorable outcomes is $E = \{2, 3, 5\}$. Thus, the number of favorable outcomes $n(E) = 3$.
Now we apply the probability formula: $P = \frac{3}{6}$.
Simplifying the fraction by dividing both the numerator and denominator by 3, we get $\frac{1}{2}$. This means there is a 50% chance of rolling a prime number.
Step 4 : Final Answer:
The probability of rolling a prime number is $\frac{1}{2}$, which is represented by option (C).