In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

Verified step by step guidance

1. <insert step 1: Understand that a die has 6 faces, numbered from 1 to 6.>
2. <insert step 2: Determine the probability of rolling a 2 on the first roll. Since there is only one '2' on a die, the probability is \( \frac{1}{6} \).>
3. <insert step 3: Determine the probability of rolling a 3 on the second roll. Similarly, the probability is \( \frac{1}{6} \).>
4. <insert step 4: Since the rolls are independent events, multiply the probabilities of each event to find the combined probability: \( \frac{1}{6} \times \frac{1}{6} \).>
5. <insert step 5: Simplify the expression to find the probability of rolling a 2 first and a 3 second.>

Key Concepts

Probability Basics

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. In the context of rolling a die, the probability of any specific outcome (like rolling a 2) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For a fair six-sided die, the probability of rolling a specific number is 1/6.

Independent Events

Two events are considered independent if the occurrence of one does not affect the occurrence of the other. In this scenario, rolling a die twice involves independent events, meaning the result of the first roll does not influence the result of the second roll. This independence allows us to multiply the probabilities of each individual event to find the overall probability.

Multiplication Rule of Probability

The multiplication rule states that the probability of two independent events both occurring is the product of their individual probabilities. For the given problem, to find the probability of rolling a 2 first and then a 3, you multiply the probability of rolling a 2 (1/6) by the probability of rolling a 3 (1/6), resulting in a combined probability of 1/36.