Find the probability of getting an even number in a single throw of a six-sided die.

Analysis

To solve this problem, we need to determine the probability of an event occurring. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space.

Concept

1. Sample Space ($S$): A standard six-sided die has six faces numbered 1, 2, 3, 4, 5, and 6. Therefore, the total number of possible outcomes $n(S) = 6$.
2. Favorable Outcomes ($E$): We are looking for an "even number." From the set $\{1, 2, 3, 4, 5, 6\}$, the even numbers are 2, 4, and 6.
3. Count of Favorable Outcomes: There are 3 even numbers, so $n(E) = 3$.
4. Probability Formula: $P(E) = \frac{n(E)}{n(S)}$

Solution

• Step 1: Identify the total outcomes: $\{1, 2, 3, 4, 5, 6\}$, so $n(S) = 6$.
• Step 2: Identify the favorable outcomes (even numbers): $\{2, 4, 6\}$, so $n(E) = 3$.
• Step 3: Apply the formula: $$P(\text{even}) = \frac{3}{6}$$
• Step 4: Simplify the fraction: $$\frac{3}{6} = \frac{1}{2}$$

Exam Tip

When dealing with dice problems, always list out your sample space if you are unsure. For a single die, the sample space is always $\{1, 2, 3, 4, 5, 6\}$. If the question asked for a prime number, remember that 1 is not a prime number, so your favorable outcomes would be $\{2, 3, 5\}$.