Problem 5 - Algebra and Trigonometry

Question

An experiment consists of rolling a die. List the elements in the following sets. (a) The sample space (b) The event "getting an even number" (c) The event "getting a number greater than 4\("\)

Step-by-Step Solution

Answer

(a) {1, 2, 3, 4, 5, 6}; (b) {2, 4, 6}; (c) {5, 6}.

Step 1: Identifying the Sample Space

A die has six faces, each with a unique number from 1 to 6. Therefore, when we roll a die, the possible outcomes are these numbers. The sample space is the set of all possible outcomes. Hence, the sample space is: \( \{1, 2, 3, 4, 5, 6\} \).

Step 2: Identifying the Event - Getting an Even Number

Even numbers are those which are divisible by 2. From the set of numbers on a die \( \{1, 2, 3, 4, 5, 6\} \), the even numbers are 2, 4, and 6. Thus, the set of outcomes for the event 'getting an even number' is: \( \{2, 4, 6\} \).

Step 3: Identifying the Event - Getting a Number Greater Than 4

We need to find numbers greater than 4 from the numbers on a die \( \{1, 2, 3, 4, 5, 6\} \). The numbers greater than 4 are 5 and 6. Therefore, the set of outcomes for this event is: \( \{5, 6\} \).

Key Concepts

Sample Space

The concept of a sample space is fundamental in probability theory. It refers to the set of all possible outcomes that can occur in a given experiment. In the case of an experiment like rolling a die, each side of the die represents a unique outcome. When we roll a die, the sample space encompasses all six possible outcomes, which are

• 1
• 2
• 3
• 4
• 5
• 6

This complete set of outcomes is denoted as \( \{1, 2, 3, 4, 5, 6\} \). Being keen on identifying the sample space aids in setting the foundation for analyzing likelihoods in probability. It helps us understand the extent and limits of the possible outcomes that can occur.

Events in Probability

An event in probability is a specific outcome or a set of outcomes within the sample space. Events are what we often seek to find probabilities for. For example, in the die-rolling experiment, consider the event of rolling an even number. To determine which outcomes constitute this event, we identify those numbers in the sample space that meet the condition of being divisible by 2.The even numbers from the set \( \{1, 2, 3, 4, 5, 6\} \) are

• 2
• 4
• 6

Thus, the event "getting an even number" is represented by the set \( \{2, 4, 6\} \). Events can be singular or contain multiple outcomes, as seen in this example, and understanding them is crucial for calculating probabilities.

Outcomes in Experiments

In the context of probability, an outcome describes a single result from an experiment. Outcomes are specific to experiments and help in identifying possible events.When considering the event "getting a number greater than 4" when rolling a die, we first check our sample space \( \{1, 2, 3, 4, 5, 6\} \) for numbers that fit this description. The outcomes greater than 4 are

• 5
• 6

Therefore, the outcomes for this event are contained in the set \( \{5, 6\} \).Recognizing outcomes allow us to develop a deeper understanding of how often these results might occur, influencing the prediction of likelihoods or probabilities in experiments.