2.5 Find unions using the addition rule | Athabasca University MATH 215: Introduction to Statistics

Athabasca University MATH 215: Introduction to Statistics

Probability

Calculate the chance that at least one of two events occurs

Probability questions often ask whether one event or another will happen. For example, a question may ask whether a randomly selected student uses the bus or the train. In probability, “or” usually includes the possibility that both events happen. The union describes this inclusive “or,” and the addition rule finds its probability without counting shared outcomes twice. This lesson defines the terms, shows how to choose and apply the rule, and explains how to report the result in context.

What you will learn

Explain the meaning of a union and an intersection.
Use the addition rule to find a union probability when events overlap.
Recognize when events are mutually exclusive and use the addition rule appropriately.
Interpret a union probability in the setting described.

1. Events, unions, and shared outcomes

An event is a collection of outcomes that meet a stated condition. If one student is selected, the event “uses the bus” consists of the outcomes in which the selected student uses the bus. A sample space is the collection of all outcomes considered in the question. Here, it could be all students in the group from which the selection is made.

A union is the event that at least one of two events occurs. If

A

means “uses the bus” and

B

means “uses the train,” the union includes students who use the bus, students who use the train, and students who use both. The symbol for the union is

A \cup B

. This is the inclusive meaning of “A or B.” If a question asks for exactly one of the events, that is a different question; a union includes the possibility of both.

The intersection is the event that both events occur. Its symbol is

A \cap B

. For the bus-and-train example, the intersection includes students who use both methods. The intersection matters because adding the number of people in

A

to the number in

B

counts everyone in the intersection twice. A union should count each person only once.

A probability is a number from

0

to

1

that describes the chance of an event. For a finite group in which each person is equally likely to be selected, the probability of an event is its count divided by the total number of people. The population is the full group of interest; a sample is the smaller group observed. A variable is the characteristic recorded, such as whether a student uses each method. A count or proportion calculated from the sample is a statistic; a number describing the full population is a parameter. In a question that asks for a random selection from an observed sample, use the sample counts and state that the probability refers to that selection.

A \cup B

A union means one event, the other event, or both.
An intersection means both events occur.
Shared outcomes belong in the union, but each is counted only once.

2. The addition rule and when to use it

Use the general addition rule when finding the probability of a union. It applies whether the events overlap or not, provided all probabilities refer to the same setting and sample space. You need the probability of each event and the probability of their intersection. If the problem gives counts instead, identify the common total and the count in both events.

The reason for subtracting the intersection is straightforward: adding the two event probabilities counts each shared outcome once as part of

A

and once as part of

B

. Subtracting the intersection once corrects this double count. Do not subtract the intersection twice; the goal is for every outcome in the union to contribute once.

Events are mutually exclusive when they cannot happen together. For example, on a single roll of one die, “the result is

1

” and “the result is

4

” cannot both be true. In this special case, the intersection has probability

0

, so the general rule reduces to adding the two probabilities. Use that shorter form only after checking that the events truly cannot overlap. Different event names, by themselves, do not mean events are mutually exclusive.

Before calculating, check that the event descriptions match what the question asks, that the probabilities or counts use the same sample space, and that any overlap value describes outcomes in both events. With counts, each count should refer to the same group and the same selection process. If the data do not give the intersection directly, do not treat it as zero unless the events are known to be mutually exclusive. The answer to a probability calculation should be between

0

and

1

; a result outside that range signals a problem with the information or arithmetic.

P(A \cup B)=P(A)+P(B)-P(A \cap B)

For overlapping events, add the probabilities and subtract the intersection once.
For mutually exclusive events, the intersection is zero and the probabilities can be added directly.
Check the shared setting and overlap before substituting values.

3. A reliable calculation and interpretation

First write down what

A

and

B

mean. Then decide whether the question asks for a union: phrases such as “A or B” and “at least one” usually signal a union. Next check for overlap. If events can occur together, use the general rule and include the intersection. If working with counts, the same idea can be used directly on counts: add the two event counts, subtract the count in both, and divide by the total to obtain a probability.

A useful algebra reminder is that division by a common total can be applied to each count: for example,

(a+b-c)/n=a/n+b/n-c/n

, when the same positive total

n

is used throughout. This is why converting each count to a probability and then applying the rule gives the same result as finding the union count first and dividing once.

After calculating, check that the union count is not larger than the total and that the probability is in range. Interpret the answer in the original setting. A probability of

0.775

means a 77.5% chance for the stated selection process; it does not promise that exactly

77.5

out of every

100

selections will have that result. Use the context and the unit, such as students or people, when explaining a count.

Name the events, check overlap, substitute, calculate, and interpret.
Use a single total when converting counts to probabilities.
A probability describes chance in the stated setting; it is not a guaranteed count.

Worked example

Bus or train use in a student sample

In a sample of 120 students, 68 report using the bus, 45 report using the train, and 20 report using both. If one student is selected at random from this sample, find the probability that the student uses the bus or the train.

Define the events
Let $A$ be the event that the selected student uses the bus, and let $B$ be the event that the student uses the train. The sample space is the 120 students in the sample, and the recorded variables are whether each student uses each method. The question asks for a union because “or” includes students who use both.
$A \cup B$
Check the overlap and conditions
The events are not mutually exclusive because 20 students use both. The event counts and the total all refer to the same sample, and the overlap is explicitly given, so the general addition rule applies. Convert the counts to probabilities using the common total of 120.
$P(A)=\frac{68}{120},\quad P(B)=\frac{45}{120},\quad P(A \cap B)=\frac{20}{120}$
Substitute into the rule
Add the probabilities of using each method, then subtract the probability of using both. The subtraction removes the students who were included in both event counts.
$P(A \cup B)=\frac{68}{120}+\frac{45}{120}-\frac{20}{120}$
Calculate and interpret
The union count is 93 students. Dividing it by the total gives the probability. Keep six decimal places in the intermediate proportion, then report the final probability to three decimal places. This is the chance that a randomly selected student from this sample uses at least one of the two methods.
$\frac{68+45-20}{120}=\frac{93}{120}=0.775000$

Answer: The probability is

0.775

, or 77.5%.

Check: Adding the bus and train counts gives 113, but the 20 students who use both are counted twice in that sum. Subtracting them once gives 93 students who use at least one method, which is no more than the sample total of 120.

Common mistakes and how to avoid them

Adding event probabilities when the events overlap, without accounting for the intersection.

Correction: Use the general addition rule and subtract the probability of the intersection once. The intersection was counted twice in the initial sum.

Reading “or” as “exactly one event.”

Correction: A union includes outcomes in both events. Use it for inclusive “or” or “at least one,” unless the question specifically says exactly one.

Assuming events are mutually exclusive because they have different descriptions.

Correction: Check whether both conditions can be true for the same outcome. If so, they overlap and the intersection must be included.

Subtracting an overlap that does not refer to the same group or subtracting it twice.

Correction: Use the intersection for the same sample space and subtract it once, because that is the amount initially counted twice.

Lesson summary

A union includes outcomes in either event or in both.
For possibly overlapping events, use $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ .
Mutually exclusive events have an intersection probability of zero, so the general rule becomes addition.
With counts, use one common total and subtract the overlap once before interpreting the probability in context.

Check your understanding

Question 1

In a group, 30 people attend a morning class, 25 attend an afternoon class, and 10 attend both. How many attend at least one class?

55
45
35
65

Show answer and explanation

45

Add the attendance counts and subtract the people counted in both:

30+25-10=45

. The answer is 45 people.

Question 2

Events

A

and

B

cannot occur together. If

P(A)=0.18

and

P(B)=0.27

, what is

P(A \cup B)

?

$0.045$
$0.18$
$0.45$
$0.63$

Show answer and explanation

0.45

Because the events cannot occur together, their intersection probability is zero. Add their probabilities:

0.18+0.27=0.45

.

Question 3

Suppose

P(A)=0.50

,

P(B)=0.40

, and

P(A \cap B)=0.15

. What is

P(A \cup B)

?

$0.90$
$0.75$
$0.35$
$0.15$

Show answer and explanation

0.75

Apply the general addition rule and subtract the overlap once:

0.50+0.40-0.15=0.75

.

Key terms

Event: A collection of outcomes that meet a condition.
Sample space: All outcomes considered in a probability question.
Union: The event that at least one of two events occurs, including the possibility that both occur.
Intersection: The event that both of two events occur.
Mutually exclusive events: Events that cannot occur together.
Probability: A number from zero to one describing the chance of an event.
Statistic: A number calculated from a sample.
Parameter: A number describing a population.

Continue through MATH 215

View the complete MATH 215 Athabasca University MATH 215: Introduction to Statistics curriculum and lessons

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 2.5. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

2.5 · Find unions using the addition rule

Athabasca University MATH 215: Introduction to Statistics

Calculate the chance that at least one of two events occurs

What you will learn

1. Events, unions, and shared outcomes

2. The addition rule and when to use it

3. A reliable calculation and interpretation

Bus or train use in a student sample

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MATH 215

About this lesson