Problem 40 - Precalculus with Limits

Question

In Exercises 39 - 42, you are given the probability that an event will not happen. Find the probability that the event will happen. \( P(E') = 0.92 \)

Step-by-Step Solution

Answer

The probability that the event will happen is 0.08.

Step 1: Understand and Identify

In probability, the sum of the probabilities of an event and its complement equals 1. This is represented as \( P(E) + P(E') = 1 \) where \( P(E) \) is the probability of event E happening, and \( P(E') \) is the probability of event E not happening.

Step 2: Substitute the Given Value

The given value is \( P(E') = 0.92 \). We need to find \( P(E) \). We can rearrange the equation from Step 1 to solve for \( P(E) \): \( P(E) = 1 - P(E') \). Now plug in the value of \( P(E') \) into this equation.

Step 3: Calculate

Substitute \( P(E') = 0.92 \) into equation from Step 2 to find \( P(E) \): \( P(E) = 1 - 0.92 = 0.08 \).

Key Concepts

Complementary Events

In probability theory, the concept of complementary events is a fundamental idea that helps us understand the total likelihood of occurrences in any event scenario.
Complementary events refer to a situation where there are only two possible outcomes: the event happening and the event not happening. For these types of events, the sum of their probabilities always equals 1.
This relationship is expressed using the formula \[ P(E) + P(E') = 1 \]where:

• \( P(E) \) is the probability of the event occurring, and
• \( P(E') \) is the probability of the event not occurring.

In practical terms, if you know one of the probabilities, you can easily find the other by subtracting the known probability from 1. For example, if the probability an event does not happen, \( P(E') \), is 0.92, then the probability of the event happening, \( P(E) \), is \( 1 - 0.92 = 0.08 \).
This simplification helps in quickly assessing the likelihood of an event when its complementary probability is already known.

Probability Calculation

Calculating probability involves understanding the likelihood of an event occurring.
To determine the probability of an event, such as flipping a coin or drawing a card, you would typically multiply the total successful outcomes by the total possible outcomes. However, the process can differ slightly for complementary events.
Given a complementary event probability of 0.92 that an event will not occur, we can find the probability of the event occurrence. Here's how:

• First, recognize that you're dealing with complementary events. The probability formula is \( P(E) = 1 - P(E') \).
• Then substitute the given complement probability: \( P(E) = 1 - 0.92 \).
• Finally, perform the subtraction: \( P(E) = 0.08 \).

This straightforward subtraction tells us the probability that the event happens. It's a handy equation when you know how likely it is for something to not happen.

Probability Theory

Probability theory forms the backbone of understanding random events. It is a mathematical framework for determining the chance of occurrence of different outcomes.
At its core, probability theory uses numerical values to express the likelihood of events. The probabilities range between 0 and 1, where:

• 0 indicates impossibility, and
• 1 indicates certainty.

This theory includes concepts of dependent events, independent events, and complementary events, among others. Each of these concepts helps us make predictions based on available data.
Consider complementary events within probability theory, like knowing the probability of rain today. If the weather forecast suggests 0.92 probability for dryness (no rain), you immediately know there's an 0.08 probability of rain.
Applying probability theory goes beyond just simple event predictions; it's used in fields like finance, science, and even sports analytics to make informed decisions based on quantifiable risks.