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 event is \( P(A^{c}) = 1 - P(A) \).

Step 1: Define the Sample Space and Events

The sample space, denoted as \( S \), is the set of all possible outcomes of an experiment. An event \( A \) is any subset of \( S \). The complement of an event \( A \), denoted as \( A^{c} \), consists of all outcomes in \( S \) that are not in \( A \).

Step 2: Apply Total Probability

The total probability of the sample space \( S \) is always 1, i.e., \( P(S) = 1 \). Events \( A \) and \( A^{c} \) together cover the entire sample space without overlap, meaning \( A \cup A^{c}= S \) and \( A \cap A^{c} = \emptyset \).

Step 3: Use Probability Addition Rule

The probability of the union of two mutually exclusive events is the sum of their probabilities: \( P(A \cup A^{c}) = P(A) + P(A^{c}) \). Since \( A \cup A^{c} = S \), we have \( P(S) = P(A) + P(A^{c}) \).

Step 4: Solve for the Complement's Probability

Given that \( P(S) = 1 \), from Step 3, we substitute: \( 1 = P(A) + P(A^{c}) \). We can rearrange to show \( P(A^{c}) = 1 - P(A) \). Thus, the probability of the complement is equal to one minus the probability of the event.

Key Concepts

Sample Space

In probability theory, the "sample space" is a foundational concept. It represents the set of all possible outcomes that can occur in a particular experiment. Imagine flipping a coin — the sample space here includes two outcomes: heads and tails.

• Denoted by the symbol \( S \), it encompasses every possible result.
• This set is crucial because it defines the range within which probabilities are calculated.

For any event you consider, it is simply a subset of this larger sample space. Understanding the sample space helps you determine how likely an event is within the broader context of all outcomes. Remember that \( P(S) = 1 \), highlighting that the sample space accounts for 100% of the potential results.

Complement of an Event

The "complement of an event" refers to everything within the sample space that does not contribute to that event. Simply put, if you have an event \( A \), its complement, noted as \( A^c \), includes all outcomes in the sample space \( S \) that are not part of \( A \).

• The idea here is to account for all possibilities: either something happens (event \( A \)), or it doesn’t (event \( A^c \)).
• Mathematically, complements help us determine probabilities of events indirectly. If you know the probability of an event, the complement's probability is just what's left: \( P(A^c) = 1 - P(A) \).

Considering complements is vital because it reinforces that probabilities cumulatively add up to the certainty of one, or a full sample space. This idea is at the heart of understanding events and their probabilities.

Mutually Exclusive Events

When we discuss "mutually exclusive events," we're referring to events that cannot happen at the same time. Think of rolling a single die — getting a 1 and getting a 2 are mutually exclusive outcomes because both cannot occur simultaneously.

• If two events \( A \) and \( B \) are mutually exclusive, then \( A \cap B = \emptyset \)—they share no outcomes.
• The probability rule for these events is straightforward: \( P(A \cup B) = P(A) + P(B) \) because there's no overlap or double-counting needed.

This concept is fundamental in probability because it simplifies how you calculate the likelihood of one of several potential events occurring. Ensuring events are mutually exclusive helps maintain accuracy in your probability assessments.