Problem 37 - 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. $$P\left(E^{\prime}\right)=0.12$$

Step-by-Step Solution

Answer

The probability that the event will happen is \(P(E) = 1 - 0.12 = 0.88\).

Step 1: Understand the variables

Here, \(P(E')\) represents the probability of the event not happening. This value is given as 0.12.

Step 2: Apply the Property of Probabilities

The probability of an event plus the probability of its complement equals one i.e., \(P(E) + P(E') = 1\).

Step 3: Find Probability of the Event

Rearranging Step 2, we find the probability of the event happening: \(P(E) = 1 - P(E')\). Substitute \(P(E') = 0.12\) into the equation to get \(P(E)\).

Key Concepts

Complement of an Event

In probability theory, understanding the concept of the complement of an event is fundamental. Imagine tossing a coin; there are two possible outcomes, heads or tails. If we define our event E as getting a head, then the complement of the event, denoted as E' or sometimes E^c, would be getting a tails.

Essentially, the complement of an event includes all outcomes in our sample space that are not part of the event E. It's like looking at the opposite side of what you're interested in. The important thing to remember is that the probability of an event E happening and the probability of its complement E' happening always add up to 1. This relationship is neatly captured by the formula:
\[ P(E) + P(E') = 1 \].

So, in our exercise, given that P(E') is 0.12 (the probability that the event does not happen), we can deduce that the probability of the event happening is simply one minus the probability of its occurrence. This relationship allows us to find missing probabilities when we know one part of the pair (the event or its complement).

Property of Probabilities

Probability theory rests on a set of axioms or properties that dictate how probabilities should behave. One such property of probabilities asserts that the probability of any event ranges from 0 to 1. A probability of 0 indicates an impossible event, and a probability of 1 signifies a certain event. Additionally, as we've seen with the complement rule, the sum of the probabilities for an event and its complement is always 1.

This property is what made it possible, in our exercise, to figure out the probability of the event occurring. Since we knew that the probability of the complement is 0.12, and considering that the sum of both must equal one, we could confidently say that the event's probability was 0.88 or \( 1 - 0.12 \).

Another vital property to understand is that the sum of the probabilities of all possible outcomes of a random experiment is 1. This foundational idea ensures a consistent and coherent framework for computing probabilities across different scenarios.

Probability Theory

Probability theory is the branch of mathematics concerned with quantifying the likelihood of different events occurring. It serves as a fundamental tool in fields ranging from statistics and finance to engineering and science. At its core, it deals with calculating the chances of various outcomes based on a defined sample space – the set of all possible outcomes of a random process.

In the example we have encountered, the process is simplified to a binary outcome situation – the event happens, or it does not. However, probability theory can become incredibly complex involving multiple events, interdependencies, conditional probabilities, and the use of combinations and permutations. This complexity is why the foundational properties, like the complement rule and the range of probabilities, are so critical. They provide a starting point to unravel more complicated probability puzzles.

Moreover, probability theory lays the groundwork for inferential statistics, where the goal is to make predictions or inferences about a population based on sample data. By understanding the probability of certain events, one can make informed decisions in the presence of uncertainty.