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 single die has 6 faces, each roll has 6 possible outcomes: 1, 2, 3, 4, 5, and 6.
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: \(\frac{3}{6} \times \frac{4}{6}\).

Key Concepts

Probability of Independent Events

When two events occur independently, the probability of both happening is the product of their individual probabilities. Rolling a die twice are independent events, so multiply the probability of the first event by the probability of the second.

Sample Space of a Die Roll

A single roll of a fair six-sided die has six equally likely outcomes: 1 through 6. Understanding this sample space helps determine the probability of specific events, such as rolling an even number or a number greater than 2.

Event Definition and Counting Favorable Outcomes

To find the probability of an event, identify all outcomes that satisfy the event's condition. For example, even numbers on a die are 2, 4, and 6; numbers greater than 2 are 3, 4, 5, and 6. Counting these helps calculate the event's probability.