Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

You are given the probability that an event will not happen. Find the probability that the event will happen. $$P\left(E^{\prime}\right)=0.84$$

Step-by-Step Solution

Answer

The probability that the event will happen is 0.16.

Step 1: Understand the problem

We are given the probability of the event not happening, represented as \( P(E')= 0.84 \). We need to find the probability of the event happening, represented as \( P(E) \). In probability, the sum of an event and its complement is always 1. Therefore, we can express the probability of the event happening as \( P(E) = 1 - P(E') \).

Step 2: Substitute the given value

Substitute \( P(E') = 0.84 \) into the formula \( P(E) = 1 - P(E') \) and solve for \( P(E) \).

Step 3: Solve for \( P(E) \)

Calculating \( P(E) = 1 - 0.84 = 0.16 \). Therefore, the probability that the event will happen is 0.16.

Key Concepts

Complementary Events

In probability, events can be categorized as either happening or not happening. When we talk about complementary events, we are referring to these two possibilities.

For any event \( E \), there is a complementary event, \( E' \), which represents the situation where the event does not occur.

• If \( P(E) \) denotes the probability of the event happening, then \( P(E') \) is the probability of it not happening.
• The relationship between an event and its complement is given by the equation: \[ P(E) + P(E') = 1 \]

This means that the sum of all possible outcomes—either the event occurs or it doesn't—equals one. Understanding this concept allows us to easily find the probability of an event if we know the probability of its complement. For example, if the probability of \( E' \) is 0.84, then \( P(E) = 1 - 0.84 = 0.16 \).

Through this simple formula, we can efficiently calculate the likelihood of an event when given its complementary probability.

Probability Theory

Probability theory is the branch of mathematics that deals with uncertainty. It provides us with the tools and methods to quantify the likelihood of different outcomes.

At its core, probability theory involves understanding how likely an event is to occur based on certain conditions or evidence. There are some key principles:

• Probability Values: Probabilities range from 0 to 1, where 0 means the event cannot happen and 1 means it's certain to happen.
• Sum of Probabilities: For any situation, the sum of probabilities of all possible outcomes is 1.
• Equally Likely Outcomes: If all outcomes are equally likely, the probability can be found by dividing the number of favorable outcomes by the total number of outcomes.

Probability theory helps us make predictions, assess risks, and make informed decisions based on incomplete data. For the given exercise, the concept is used to determine the likelihood of an event simply by subtracting its complement from one.

Event Outcomes

Event outcomes refer to the possible results that can arise from an experiment or situation. When dealing with probability, each outcome has a specific likelihood of occurring.

To understand event outcomes more thoroughly, consider the following:

• Sample Space: The set of all possible outcomes of an experiment, often denoted by \( S \).
• Favorable Outcomes: The outcomes that match the event of interest.
• Complementary Outcomes: These are the outcomes that do not match the event of interest.

In probability problems, knowing whether an outcome is part of the event or its complement helps in calculating probabilities accurately. For the problem in question, the outcome of the event not happening has a probability of 0.84. This means that the event happening is simply the complementary outcome, with a probability of 0.16 (since 1 - 0.84 = 0.16).

Understanding these outcomes allows us to better analyze situations and anticipate results in real-world scenarios.