Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that if \(A^{c}\) is the complement of \(A\), that is, the set of all outcomes in the sample space \(S\) that are not in \(A\), then \(P\left(A^{c}\right)=1-P(A)\).

Step-by-Step Solution

Answer

The probability of the complement, \(P(A^c)\), equals 1 minus the probability of the set, \(P(A)\).

Step 1: Understanding Complement Definition

The complement of a set \(A\), denoted \(A^c\), comprises all elements in the sample space \(S\) that are not in \(A\). Thus, the union of \(A\) and \(A^c\) is the entire sample space: \(A \cup A^c = S\).

Step 2: Applying Probability Axiom for Sample Space

One of the axioms of probability states that the probability of the entire sample space \(S\) is equal to 1: \(P(S) = 1\).

Step 3: Probability of Union for Disjoint Sets

Since \(A\) and \(A^c\) are disjoint sets (they have no elements in common), the probability of their union is the sum of their probabilities: \(P(A \cup A^c) = P(A) + P(A^c)\).

Step 4: Substitute Probability of Sample Space

From the previous steps, substitute \(P(S)\) with 1: \(P(A) + P(A^c) = 1\).

Step 5: Isolate Probability of Complement

Rearranging the equation \(P(A) + P(A^c) = 1\) to solve for \(P(A^c)\), we get \(P(A^c) = 1 - P(A)\).

Key Concepts

Complement of a Set

In the realm of probability theory, the complement of a set captures the concept of outcomes that are not part of a designated set. Let's break it down with a simple example. If we consider a set \( A \) within a sample space \( S \), the complement, denoted as \( A^c \), consists of everything in \( S \) that is not in \( A \). This means:

• The elements in \( A \) and \( A^c \) together make up the entire sample space \( S \).
• The formal notation for this is \( A \cup A^c = S \).
• The sets \( A \) and \( A^c \) are mutually exclusive (they do not share any elements).

This feature of complements is crucial in various probability calculations because it allows you to consider what does not happen, sometimes simplifying complex problems. For example, if you know the probability of an event, you automatically know the probability of it not occurring by considering its complement.

Sample Space

The sample space is a foundational element in probability theory. It encompasses every possible outcome of a particular experiment or random trial. Let's imagine you have a six-sided die. The sample space \( S \) for rolling the die would be \( \{1, 2, 3, 4, 5, 6\} \). Here’s what’s important about sample spaces:

• All elements in a sample space are mutually exclusive, meaning no two outcomes can happen at the same time.
• The probability of the sample space, \( P(S) \), is always 1. This represents certainty that one of the possible outcomes will occur.
• Sample spaces pave the way for defining events, which are subsets of the sample space.

Understanding the sample space helps in calculating probabilities of events within it. It sets the stage for determining all possible outcomes and how likely each event is to occur.

Probability Axioms

Probability axioms form the backbone of probability theory. They are the basic rules or principles that all probabilities follow, ensuring consistency and logic within the theory. There are three main axioms to consider:

• Non-negativity: The probability of any event is a non-negative number. Mathematically, this means \( P(A) \geq 0 \) for any event \( A \) within the sample space \( S \).
• Normalization: The probability of the entire sample space is equal to one: \( P(S) = 1 \). This axiom ensures that something in the sample space is bound to happen.
• Additivity: For any two disjoint events \( A \) and \( B \), the probability of their union is the sum of their individual probabilities: \( P(A \cup B) = P(A) + P(B) \).

These axioms are fundamental in calculating and interpreting probabilities of events. They ensure any combination of events adheres to a logical pattern. Moreover, when using probabilities in real-world applications, these axioms guide us to make sound assumptions and predictions.