Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

The complement of an event \(A\) is the collection of all outcomes in the sample space that are not in \(A\). If the probability of \(A\) is \(P(A),\) then the probability of the complement \(A^{\prime}\) is given by \(P\left(A^{\prime}\right)=1-P(A) .\) You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=0.36$$

Step-by-Step Solution

Answer

The probability that the event will not happen, \(P(E')\), is \(0.64\).

Step 1: Identify the given probability

The given probability of event \(E\) happening, denoted as \(P(E)\), is \(0.36\).

Step 2: Apply the formula for the complement

The probability of the complement of the event \(E'\) is given by the formula \(P(E') = 1 - P(E)\). Substituting the given value, we have \(P(E') = 1 - 0.36\).

Step 3: Calculate the result

Subtracting \(0.36\) from \(1\) gives us the probability of event \(E\) not happening, \(P(E') = 0.64\).

Key Concepts

Complement of an Event

In probability theory, the complement of an event is a fundamental concept. If you have an event, say \(A\), its complement, denoted as \(A'\), consists of everything in the sample space that is not part of \(A\). Essentially, it's the opposite of the event you're interested in.
When knowing the probability of an event occurring, you can easily find the probability of it not occurring by using the complement rule. This rule is expressed as:

• \(P(A') = 1 - P(A)\)

This formula is handy because probabilities of all possible outcomes must add up to 1. So if you know \(P(A)\), you subtract it from 1 to find \(P(A')\).
For example, if the probability of an event happening, \(P(E)\), is 0.36, then the probability of it not happening, \(P(E')\), is 1 - 0.36 = 0.64.
Understanding complements helps in complex problems where multiple scenarios or events are involved.

Sample Space

The sample space is a key concept in probability theory, referring to the set of all possible outcomes of a particular experiment or random trial. Notably, it contains every outcome, forming the foundation on which probability calculations are built.
Sample spaces can be simple or complex.
For example:

• In a coin toss, the sample space is \(\{heads, tails\}\).
• Rolling a six-sided die has a sample space of \(\{1, 2, 3, 4, 5, 6\}\).

Each outcome must be equally likely for calculating probabilities. Understanding the sample space is crucial because it helps ensure that all potential outcomes are considered.
This is particularly important in problems involving calculations of event probabilities and their complements, as you need to know the entirety of possible results to accurately apply the probability formula.

Probability Formula

A probability formula is a tool used to calculate the likelihood of an event occurring. It's a core part of any probability problem and research. The basic formula to find the probability of an event \(A\), represented as \(P(A)\), is defined by:

• \(P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in the sample space}}\)

This formula provides a proportion that expresses how likely an event is to occur.
Another common formula is for the complement, which we use to find the probability of an event not occurring:

• \(P(A') = 1 - P(A)\)

By using these formulas, students can solve a variety of probability problems.
For instance, if you know \(P(E) = 0.36\), then \(P(E')\) is computed as \(1 - P(E) = 0.64\). Understanding these formulas is central for conducting further calculations in probabilities.