Problem 36 - Precalculus with Limits

Question

In Exercises 35 - 38, 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 of the event not happening is \(0.64.\)

Step 1: Understanding the problem

The probability of an event happening is given as \( P(E) = 0.36 \). To find the probability that the event does not happen, we need to subtract this given probability from 1.

Step 2: Calculate the probability

subtract the given probability from 1. This is done as follows: \( P(\text{{not }} E) = 1 - P(E) = 1 - 0.36. \)

Key Concepts

Complementary Events

In probability theory, every event has a complementary counterpart. Simply put, if an event happens, its complement does not. Conversely, if an event does not happen, its complement happens. These two outcomes cover all possibilities and together they form the entire sample space. For any event with probability \( P(E) \), its complementary event \( \text{not } E \) represents the probability of the event not occurring. This can be calculated using the formula:

• \( P(\text{not } E) = 1 - P(E) \)

This relationship ensures that \( P(E) + P(\text{not } E) = 1 \). This means that the sum of the probabilities of an event and its complement must equal 1.
This principle of complementary events is fundamental in probability because it helps us determine probabilities efficiently. If you know one, you can find the other with ease.

Event Probability

When discussing probability, we often refer to the likelihood of a specific event occurring within a well-defined scenario known as the sample space. Event probability is a measure of this likelihood. It's a value between 0 and 1, where 0 indicates impossible, and 1 means certain.
For example, if the probability of an event \( E \) is given as \( P(E) = 0.36 \), it means there is a 36% chance of the event occurring. Understanding how to interpret these values is crucial for evaluating risks and making decisions based on probability.

• A probability of 0 means the event cannot occur.
• A probability of 1 means the event will definitely occur.
• Probabilities between 0 and 1 indicate varying chances of an event happening.

Calculating probabilities accurately involves considering all possible outcomes and determining their likelihood within the system.

Basic Probability Rules

Probability relies on a few simple yet powerful rules. These rules help in calculating the probabilities of different events consistently and accurately. Some basic rules to remember include:

• The sum of probabilities of all possible outcomes always equals 1.
• The probability of an event occurring and its complementary event equals 1 (i.e., \( P(E) + P(\text{not } E) = 1 \)).
• If events are mutually exclusive, the probability of either event occurring is the sum of their individual probabilities.

These foundational rules ensure proper understanding and calculation of probabilities. They apply to a variety of scenarios and are essential in more complex probability calculations. By mastering these rules, you’ll be better equipped to tackle any probability problem you encounter.