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.
31. Rolling a pair of six-sided dice

Verified step by step guidance

1. Define the probability experiment: Rolling a pair of six-sided dice means each die has six faces numbered from 1 to 6, and both dice are rolled simultaneously.
2. Identify the sample space: The sample space consists of all possible ordered pairs (x, y), where x represents the outcome of the first die and y represents the outcome of the second die. Since each die has 6 outcomes, there are 6 × 6 = 36 possible outcomes.
3. List the sample space: The sample space can be written as S = {(1,1), (1,2), (1,3), ..., (6,6)}, where each pair represents the result of the first die and the second die.
4. Determine the number of outcomes: Count the total number of pairs in the sample space. Since there are 6 outcomes for the first die and 6 outcomes for the second die, the total number of outcomes is 6 × 6 = 36.
5. Draw a tree diagram: Start with the first die, branching out to its 6 possible outcomes (1, 2, 3, 4, 5, 6). From each of these branches, create 6 additional branches for the outcomes of the second die. This will visually represent all 36 possible outcomes.

Key Concepts

Sample Space

The sample space is the set of all possible outcomes of a probability experiment. In the context of rolling a pair of six-sided dice, the sample space includes every combination of the two dice, which can range from (1,1) to (6,6). Understanding the sample space is crucial for calculating probabilities and analyzing outcomes.

Outcomes

An outcome is a specific result of a probability experiment. When rolling two dice, each combination of the numbers shown on the dice represents a unique outcome. The total number of outcomes in this experiment can be calculated by multiplying the number of faces on each die, which is 6 x 6 = 36 outcomes.

Tree Diagram

A tree diagram is a visual representation used to illustrate all possible outcomes of a probability experiment. For rolling two dice, a tree diagram can show the first die's outcomes branching into the second die's outcomes, helping to visualize the sample space and count the total outcomes systematically. This tool is particularly useful for complex experiments with multiple stages.