Problem 12 - Calculus for Biology and Medicine

Question

In Problems \(9-12\), assume that $$ \Omega=\\{1,2,3,4,5\\} $$ \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\). Find \(P(A \cup B)\).

Step-by-Step Solution

Answer

The probability \(P(A \cup B) = 1.0\).

Step 1: Identify Elements in A and B

Recall the sets given: - Set \( A = \{1, 3, 5\} \) - Set \( B = \{2, 3, 4\} \). We need to find the union of these two sets, \( A \cup B \), which includes all elements that are in \( A \), \( B \), or both.

Step 2: Find the Union A ∪ B

The union of sets \( A \) and \( B \) consists of all unique elements from both sets. Thus, we have:\[ A \cup B = \{1, 2, 3, 4, 5\} \] each element present in either \( A \) or \( B \) is included.

Step 3: Determine the Probability of Each Element in A ∪ B

We need to list the probabilities of each element in \( A \cup B \):- \( P(1) = 0.1 \)- \( P(2) = 0.2 \)- \( P(3) = 0.05 \)- \( P(4) = 0.05 \)- We assume \( P(5) \) must be defined to complete \( \Omega \)'s total 1 probability. Since no probability is given, it can be computed as:\[ P(5) = 1 - (P(1) + P(2) + P(3) + P(4)) = 1 - (0.1 + 0.2 + 0.05 + 0.05) = 0.6 \].

Step 4: Calculate P(A ∪ B)

We now find \( P(A \cup B) \) by summing the probabilities of all elements in the union:\[P(A \cup B) = P(1) + P(2) + P(3) + P(4) + P(5) = 0.1 + 0.2 + 0.05 + 0.05 + 0.6 = 1.0.\]

Key Concepts

Set Theory

Set theory is a mathematical concept that deals with the collection of objects. These objects are often referred to as "elements" or "members" of the set. Let's break down some basic ideas:

• **Sets** are written with curly brackets and can contain numbers, symbols, or other sets. For example, the set containing the numbers 1, 3, and 5 would be written as \( \{1, 3, 5\} \).
• Set theory is foundational in mathematics, particularly in probability, as it provides a clear language to discuss collections of objects.
• **Equality of sets**: Two sets are equal if they contain the same elements, regardless of the order. For example, \( \{1, 3, 5\} \) is equal to \( \{5, 3, 1\} \).
• We often represent the universal set, denoted \( \Omega \), as the set of all possible outcomes. It includes every element under consideration.

Understanding these basic concepts of set theory helps in tackling more complex problems involving probability and logical operations on sets.

Union of Sets

The union of sets is a fundamental operation in set theory, symbolized by \( \cup \). When you take the union of two sets, you combine all their elements into a new set, without duplicating any elements.

For instance, if we have the sets \( A = \{1, 3, 5\} \) and \( B = \{2, 3, 4\} \), the union \( A \cup B \) would be \( \{1, 2, 3, 4, 5\} \). Notice how each number appears only once in the union set, despite \( 3 \) being present in both sets A and B.

• In combining the elements of \( A \) and \( B \), one must ensure no element repeats in \( A \cup B \).
• The concept of union is widely used in various fields: computer science, database management, and of course, probability theory.
• Visualizing sets as overlapping circles in diagrams, often called Venn diagrams, can help grasp the idea of union.

The union provides a way to consider all possible outcomes when dealing with distinct sets of data, integral for understanding probability.

Probability Calculation

Probability calculation involves assigning a numerical value to how likely an event is to occur. When dealing with events expressed as sets, understanding their union - like \( A \cup B \) - helps in determining the probability of any of the events occurring.

To calculate the probability of a union of sets, you would add together the probabilities of all distinct elements within the union. Let's break it down with our example:

• The probability of each element reflects how likely that particular outcome is, with all probabilities in the universal set \( \Omega \) summing to 1.
• The given probabilities were \( P(1)=0.1 \), \( P(2)=0.2 \), \( P(3)=P(4)=0.05 \). The missing probability \( P(5) \) was calculated as \( 1 - (0.1 + 0.2 + 0.05 + 0.05) = 0.6 \).
• Therefore, the probability of the union \( A \cup B \) was \( 0.1 + 0.2 + 0.05 + 0.05 + 0.6 = 1.0 \).

Probability calculation is crucial in decision-making processes, risk assessment, and predictions, making it an essential concept in many real-world applications.