In Exercises 29-32, find the probability.
31. A 12-sided die, numbered 1 to 12, is rolled. Find the probability that the roll results in an odd number or a number less than 4.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability of rolling an odd number OR a number less than 4 on a 12-sided die. This involves calculating the probability of two events and combining them using the formula for the union of two events.
2. Step 2: Define the events. Let Event A represent rolling an odd number, and Event B represent rolling a number less than 4. The die is numbered from 1 to 12, so the sample space consists of these 12 outcomes.
3. Step 3: Identify the outcomes for each event. For Event A (odd numbers), the outcomes are {1, 3, 5, 7, 9, 11}. For Event B (numbers less than 4), the outcomes are {1, 2, 3}.
4. Step 4: Determine the union of the two events. The union includes all outcomes that are either odd or less than 4. Combine the outcomes from Event A and Event B, ensuring no duplicates. The union is {1, 2, 3, 5, 7, 9, 11}.
5. Step 5: Calculate the probability. The probability of an event is the number of favorable outcomes divided by the total number of outcomes in the sample space. Count the outcomes in the union (7 outcomes) and divide by the total number of outcomes (12 outcomes). Use the formula: \( P(A \cup B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \).

Key Concepts

Probability

Probability is a measure of the likelihood that a particular 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 this context, we calculate the probability of rolling an odd number or a number less than 4 on a 12-sided die.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For a 12-sided die, the sample space consists of the numbers 1 through 12. Understanding the sample space is crucial for calculating probabilities, as it provides the total number of outcomes against which specific events can be measured.

Union of Events

The union of events refers to the occurrence of at least one of the events in a set. In this case, we are interested in the union of rolling an odd number and rolling a number less than 4. To find the probability of the union, we must consider the individual probabilities of each event and subtract any overlap to avoid double counting.