Identifying the Sample Space of a Probability Experiment In Exercises 25-32, identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate.
32. Rolling a six-sided die, tossing two coins, and spinning the fair spinner shown

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Step 1: Understand the components of the probability experiment. The experiment involves rolling a six-sided die, tossing two coins, and spinning a fair spinner with three sections labeled 1, 2, and 3.

Step 2: Identify the sample space for each individual component. For the six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. For the two coins, each coin has two possible outcomes: {Heads, Tails}, resulting in a combined sample space of {HH, HT, TH, TT}. For the spinner, the sample space is {1, 2, 3}.

Step 3: Combine the sample spaces of all components to form the overall sample space. Since the outcomes of the die, coins, and spinner are independent, the total sample space is the Cartesian product of the individual sample spaces.

Step 4: Calculate the total number of outcomes in the sample space. Multiply the number of outcomes for each component: 6 outcomes for the die × 4 outcomes for the coins × 3 outcomes for the spinner = 72 total outcomes.

Step 5: Draw a tree diagram to visually represent the sample space. Start with the outcomes of the die, branch out to the outcomes of the coins, and then branch further to the outcomes of the spinner. This will help illustrate all possible combinations of outcomes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Space

The sample space of a probability experiment is the set of all possible outcomes. For the given experiment of rolling a die, tossing two coins, and spinning a spinner, the sample space includes every combination of results from these actions. Understanding the sample space is crucial for calculating probabilities and analyzing the experiment.

Probability

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. In this context, it can be calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space. Knowing how to compute probabilities helps in making informed predictions about the results of the experiment.

Tree Diagram

A tree diagram is a visual representation used to illustrate all possible outcomes of a probability experiment. Each branch of the tree represents a possible outcome from one action, leading to further branches for subsequent actions. This tool is particularly useful for organizing complex experiments, such as the one described, making it easier to identify the sample space and calculate probabilities.

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