Problem 33 - 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. $$P(E)=0.75$$

Step-by-Step Solution

Answer

The probability that the event will not happen, is 0.25.

Step 1: Understanding the problem

In this problem, the event \(E\) has a probability of 0.75. We are asked to find the probability that event \(E\) will not happen which can be denoted as \(E'\), the complement of \(E\).

Step 2: Apply the formula of Probability of Complement

The probability of the complement of an event is given by \(P(E') = 1 - P(E)\). Using the provided probability of event \(E\) (0.75), we can substitute into the formula to get \(P(E') = 1 - 0.75\).

Step 3: Calculate the probability

So calculating the expression, we get \(P(E') = 0.25\). Therefore the probability that the event will not happen is 0.25.

Key Concepts

Complementary events

In probability theory, understanding complementary events is key. If you have an event \( A \), its complement \( A' \) includes all the outcomes that are not part of \( A \). Imagine flipping a coin. If \( A \) represents landing a head, \( A' \) would be landing a tail.
This relationship is useful because the probability of an event and its complement always add up to 1. So, if you know the probability of an event occurring, you can easily find the probability of it not occurring using:

• \( P(A') = 1 - P(A) \)

To illustrate this, if an event has a 75% chance of occurring, its complement has a 25% chance of not occurring. This formula simplifies calculations and helps in scenarios where finding the complement is easier.

Sample space

The sample space is a foundational concept in probability, representing the set of all possible outcomes of an experiment. For example, when tossing a die, the sample space is \( \{1, 2, 3, 4, 5, 6\} \).
Every event is a subset of this sample space. This means that understanding sample spaces allows us to define events and their complements. Knowing the sample space also helps in visualizing the problem and ensures no outcomes are left out.

• Concrete identification of all possible outcomes.
• Clear understanding of events as subsets.

Sample spaces provide the complete picture needed to calculate probabilities accurately. They are like the "universe" of possible results for the specific context you're analyzing.

Probability formula

Probability formulas are the tools used to calculate probabilities of events. The basic probability formula determines the likelihood of an event \( E \) happening:

• \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)

In our exercise, we used the complement formula, which is a specific type applied when you are interested in the probability of an event not occurring:

• \( P(E') = 1 - P(E) \)

These formulas are crucial to solve probability problems effectively. They guide us in applying mathematics to real-world scenarios, ensuring our results are logical and accurate.