In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

Verified step by step guidance

1. Identify the sample space for each roll of the die. Since a die has 6 faces, each roll has 6 possible outcomes.
2. Determine the favorable outcomes for the first roll: rolling an even number. The even numbers on a die are 2, 4, and 6, so there are 3 favorable outcomes.
3. Determine the favorable outcomes for the second roll: rolling a number greater than 2. The numbers greater than 2 are 3, 4, 5, and 6, so there are 4 favorable outcomes.
4. Calculate the probability of each event separately. The probability of rolling an even number first is $\frac{3}{6}$, and the probability of rolling a number greater than 2 second is $\frac{4}{6}$.
5. Since the two rolls are independent events, multiply the probabilities of each event to find the combined probability: $\frac{}{}\backslash times\frac{}{}$.

Key Concepts

Sample Space of a Die Roll

The sample space is the set of all possible outcomes. For a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Understanding this helps identify which outcomes meet the conditions given in the problem.

Probability of Independent Events

When rolling a die twice, each roll is independent, meaning the outcome of the first roll does not affect the second. The probability of both events occurring is the product of their individual probabilities.

Event Definition and Counting Favorable Outcomes

Defining the events clearly is crucial: the first event is rolling an even number (2, 4, 6), and the second is rolling a number greater than 2 (3, 4, 5, 6). Counting the favorable outcomes for each event allows calculation of their probabilities.