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An exercise program at a hospital is often a structured schedule of physical activities designed to improve patients' health. In this scenario, the program consists of two types of classes – swimming and aerobics. Each class offers unique benefits:

• Swimming provides a low-impact workout. It's great for cardiovascular health and reducing joint pain.
• Aerobics boosts heart rate and enhances lung capacity while being an effective form of weight management.

Understanding which patients are enrolled in these activities allows us to explore different probability questions. For this exercise, knowing how many patients participate in at least one of these classes (132) sets the stage for calculating probabilities related to enrollment.

A Venn Diagram is a useful tool when visually representing relationships and overlaps between different sets. In this hospital exercise program, a Venn Diagram could help illustrate the distribution of the 300 patients:

• One circle for the 78 patients attending the swimming class.
• Another circle for the 84 patients attending the aerobics class.
• The intersection of these circles represents the 30 patients who are in both classes.

Such a diagram is helpful to quickly interpret which patients belong to one or both groups, allowing for simpler calculations of reach, overlap, and exclusions. By presenting information graphically, complex relationships between the groups become easier to grasp. One can easily see how changes to these classes might impact the overall program and enrollment.

Set theory provides a mathematical foundation for dealing with collections of objects, known as sets. In our exercise, we can define two sets:

• Set A: Patients who participate in the swimming class.
• Set B: Patients involved in the aerobics class.

In set theory, we are often interested in:

• The Union of sets, denoted as \(|A \cup B|\), representing all patients in either or both classes.
• The Intersection of sets, noted as \(|A \cap B|\), representing patients who attend both classes.

In this example, we use the formula, \(|A \cup B| = |A| + |B| - |A \cap B|\), to find how many patients attend both classes. Here, knowing the values for \(|A|\), \(|B|\), and \(|A \cup B|\) helps in determining \(|A \cap B|\). This application of set theory allows us to analyze and solve the problem efficiently.

In probability, the complement refers to the event that encompasses all outcomes not in the specified event. Calculating complements is a handy approach when determining probabilities of exclusion.
For instance, when we find the probability of patients not in the exercise program, we seek the complement of those in the program. This means:

• The total number of patients is 300.
• The number in the program is 132.
• Thus, the complement, patients not in the program, is calculated as 300 minus 132.

By considering all 168 patients who are not signed up for any class, we easily determine the likelihood of selecting one of these patients at random. In this exercise, the complement offers a straightforward path to shifting focus from inclusion (who is in the program) to exclusion (who isn't), granting a complete view of the participation landscape.