Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.3.7

Chapter 3, Problem 3.3.7

Graphical Analysis In Exercises 7 and 8, determine whether the events shown in the Venn diagram are mutually exclusive. Explain your reasoning.

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Step 1: Understand the concept of mutually exclusive events. Two events are mutually exclusive if they cannot occur at the same time. In other words, their intersection is empty, meaning there are no outcomes that belong to both events.

Step 2: Analyze the Venn diagram provided. The diagram shows two events: 'Wearing a polo shirt' and 'Wearing Bermuda shorts.' The sample space represents all possible clothing combinations.

Step 3: Observe the intersection of the two circles in the Venn diagram. The overlapping region indicates that there are outcomes where both events occur simultaneously, i.e., a person can wear both a polo shirt and Bermuda shorts at the same time.

Step 4: Based on the presence of the intersection, conclude that the events are not mutually exclusive. The existence of shared outcomes means the events can occur together.

Step 5: Explain the reasoning clearly: Since the Venn diagram shows an overlap between the two events, they are not mutually exclusive. Mutually exclusive events would have no overlap in the diagram.

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Key Concepts

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

Mutually Exclusive Events

Mutually exclusive events are those that cannot occur at the same time. In probability, if one event happens, the other cannot. For example, if you are wearing a polo shirt, you cannot simultaneously be wearing Bermuda shorts if they are defined as mutually exclusive options.

Venn Diagrams

Venn diagrams are graphical representations used to show the relationships between different sets. In this context, the circles represent the sets of clothing options, with overlapping areas indicating that some outcomes can belong to both sets, which is crucial for determining if the events are mutually exclusive.

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. In this case, the sample space consists of all clothing combinations, and understanding it helps in analyzing the events represented in the Venn diagram to determine their exclusivity.

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