18. Rolling a Die You roll a die. Find the probability of each event.
b. Rolling a 2 or an odd number

Verified step by step guidance

1. Step 1: Understand the problem. A standard die has 6 faces numbered 1 through 6. The goal is to find the probability of rolling a 2 or an odd number. Recall that the probability of an event is calculated as the ratio of favorable outcomes to the total number of possible outcomes.
2. Step 2: Identify the total number of possible outcomes. Since the die has 6 faces, the total number of outcomes is 6.
3. Step 3: Determine the favorable outcomes for the event 'rolling a 2 or an odd number.' The odd numbers on a die are 1, 3, and 5. Additionally, the number 2 is included in the event. Therefore, the favorable outcomes are {1, 2, 3, 5}.
4. Step 4: Count the number of favorable outcomes. The set {1, 2, 3, 5} contains 4 outcomes.
5. Step 5: Calculate the probability. Use the formula \( P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Substitute the values: \( P(E) = \frac{4}{6} \). Simplify the fraction if needed.

Key Concepts

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. An event with a probability of 0 means it cannot happen, while a probability of 1 means it is certain to happen. In the context of rolling a die, the probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For a single roll of a fair six-sided die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}. Understanding the sample space is crucial for calculating probabilities, as it provides the basis for determining how many outcomes are favorable for a given event.

Compound Events

A compound event is an event that consists of two or more simple events. In this case, rolling a 2 or an odd number involves two simple events: rolling a 2 and rolling an odd number (1, 3, or 5). To find the probability of a compound event, one must consider the individual probabilities of each simple event and account for any overlap, ensuring that outcomes are not double-counted.