Problem 36 - 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. $$P(E)=\frac{7}{8}$$

Step-by-Step Solution

Answer

The probability that the event will not happen is \(\frac{1}{8}\)

Step 1: Identify the given probability

The probability of event E, denoted by \(P(E)\), is given as \(\frac{7}{8}\).

Step 2: Calculate the complement's probability

The probability of the complement of E, denoted by \(P(E')\), is calculated using the formula \(P(E')=1-P(E)\). Substituting the given \(P(E)\) into the formula, we get \(P(E')=1-\frac{7}{8}=\frac{1}{8}\).

Key Concepts

Sample Space

The term sample space is fundamental in the study of probability. It represents the set of all possible outcomes of a probability experiment. For example, when we flip a standard coin, the sample space consists of two outcomes: {Heads, Tails}.

Understanding the sample space is crucial because it provides the context for evaluating probabilities of different events. An event is simply a subset of the sample space. For instance, if the event is 'getting a Heads', then in our earlier coin flip example, the event is one of the two possible outcomes in the sample space.

In more complex scenarios, the sample space can be quite large. Consider rolling a six-sided die: the sample space would be {1, 2, 3, 4, 5, 6}, representing each possible roll. For an event such as 'rolling an even number', the event would comprise the outcomes {2, 4, 6} within the sample space.

Complement of an Event

The complement of an event includes all the outcomes in the sample space that are not in the original event. It essentially answers the question, 'What else could happen apart from the given event?'

For instance, if we have an event 'A' that represents 'rolling a 5 on a die,' the complement of 'A' (denoted as 'A'') would include the outcomes that are not '5', which in this case would be {1, 2, 3, 4, 6}. This concept underscores an important principle in probability: the sum of the probabilities of an event and its complement always equals 1. This is because either the event or its complement must occur.

To visualize, imagine you have a pie representing the sample space; if the event 'A' takes up a slice of the pie, the complement of 'A' consists of the rest of the pie. This relationship between an event and its complement is a powerful tool for solving many probability problems.

Probability Calculation

Performing a probability calculation enables us to quantify the chance of an event occurring. The probability of any event 'E' is calculated by taking the number of favorable outcomes and dividing it by the total number of outcomes in the sample space. Mathematically, it's expressed as:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Let's apply this to our previous exercise. If we know the probability of event 'E' is \(P(E)=\frac{7}{8}\), and we want to find the probability of not 'E', which we denote as 'E'', we use the complement rule. Since the total probability must sum to 1, we subtract the probability of the event 'E' from 1 as such:
\[ P(E') = 1 - P(E) \]
Substituting the known probability of 'E' into this formula, we find the probability of 'E'' to be \(1 - \frac{7}{8} = \frac{1}{8}\), which is equivalent to saying that the chance of event 'E' not occurring is 1 out of 8.