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P2 · Solve probability problems with mutually exclusive or overlapping events

Learn to solve probability problems with mutually exclusive or overlapping events through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Probability

Using counts and set diagrams to find the probability of “A or B”

A student might ask for the probability that a randomly chosen classmate takes art or music. Some students may take both. If we add the number taking art to the number taking music, those students are counted twice. The key is to notice whether the events overlap. This lesson reviews basic probability, explains both cases, and shows how to solve problems with counts and probabilities.

What you will learn

  • Explain what mutually exclusive and overlapping events mean.
  • Use a Venn diagram to organize counts for two events.
  • Find the probability that at least one of two events occurs.
  • Choose the correct addition rule and avoid counting outcomes twice.

Review: probability and events

An outcome is one possible result of a situation. An event is a group of outcomes that match a condition. For example, when one student is selected from a class, the event “takes art” includes every student who takes art.
A sample space is the complete set of possible outcomes for the situation. When outcomes are equally likely, probability is the number of favourable outcomes divided by the total number of outcomes. A probability is between 0 and 1. A probability of 0 means an event cannot happen; a probability of 1 means it must happen.
The word “or” usually means that event A happens, event B happens, or both happen. This is called the union of the events. In this lesson, “A or B” includes the possibility that both occur.
P(A)=n(A)n(S)P(A)=\frac{n(A)}{n(S)}
  • Count favourable outcomes and divide by the total when outcomes are equally likely.
  • “A or B” includes outcomes in both events unless the question says otherwise.

Mutually exclusive events

Two events are mutually exclusive if they cannot happen at the same time. For example, on one roll of a standard six-sided die, “roll a 2” and “roll a 5” are mutually exclusive. One roll cannot show both numbers.
For mutually exclusive events, adding the probabilities does not count any outcome twice. The probability of A or B is the probability of A plus the probability of B.
Before using this rule, check what one outcome represents. Events can be mutually exclusive in one situation but not in another. “Takes art” and “takes music” are not mutually exclusive if a student can take both.
P(A or B)=P(A)+P(B)
  • Mutually exclusive events have no outcomes in common.
  • For mutually exclusive events, add their probabilities directly.

Overlapping events and Venn diagrams

Events overlap when some outcomes belong to both events. A Venn diagram shows this shared group in the intersection, the part where the two circles overlap. The region outside both circles represents outcomes in neither event.
When you add the counts or probabilities for A and B, outcomes in the intersection are included once in A and once in B. Subtract the intersection once to correct this double count. A Venn diagram is useful because it makes the shared outcomes visible before you calculate.
The same rule works with counts or with probabilities. If you use counts, divide the final favourable count by the total number of outcomes. If you use probabilities, use the probability version of the rule directly.
P(A or B)=P(A)+P(B)-P(A and B)
  • The intersection contains outcomes in both events.
  • Subtract the intersection once when finding the probability of A or B.

A reliable method

First identify the total number of equally likely outcomes, or the relevant probabilities supplied in the question. Then define events A and B in words. Decide whether they are mutually exclusive by asking whether one outcome can belong to both.
If the events are mutually exclusive, add their probabilities. If they overlap, find the shared part and subtract it once. With counts, you can place the shared count in the centre of a Venn diagram, fill the two non-overlapping regions, and then calculate the desired total.
Check that your answer is a probability between 0 and 1. If you found a count, make sure it is no greater than the total. State the result in context so it is clear what the probability describes.
  • Define the events before choosing a rule.
  • For overlapping events, identify the shared part before adding.
  • Check that a probability is between 0 and 1.

Worked example

Mutually exclusive outcomes on a spinner

A fair spinner has 8 equal sections numbered 1 to 8. What is the probability of landing on 2 or 7?
  1. Name the events
    Let A be landing on 2 and B be landing on 7. One spin gives only one number, so the events cannot happen together. They are mutually exclusive.
  2. Add the probabilities
    Each number has one of the eight equally likely sections. Since there is no overlap, add the two probabilities. P(A or B)=18\frac{1}{8}+18\frac{1}{8}
  3. State the result
    The probability of landing on either number is one quarter. P(A or B)=14\frac{1}{4}
Answer: The probability is 14\frac{1}{4}.
Check: The two favourable sections are distinct, so 2 of the 8 sections are favourable; 28=14\frac{2}{8}=\frac{1}{4}.

Worked example

Overlapping choices in a class

In a class of 40 students, 18 take art, 21 take music, and 6 take both. If one student is selected at random, what is the probability that the student takes art or music?
Art and music choices
Art and music choicesStudents in each region of the classArtMusic126157Neither

Students in each region of the class

  1. Define the events
    Let A mean that the selected student takes art, and B mean that the student takes music. Because 6 students take both, these events overlap.
  2. Find the separate regions
    The art-only group has 12 students because the 6 students taking both are part of the 18 taking art. The music-only group has 15 students for the same reason.
    18−6=12,21−6=1518-6=12,\quad 21-6=15
  3. Count students in at least one group
    Add the art-only, music-only, and both groups. This counts every student taking at least one subject exactly once.
    12+15+6=3312+15+6=33
  4. Convert the count to probability
    There are 40 students in total, and 33 take art or music. The selection is random, so the probability is the favourable count divided by the total. P(A or B)=3340\frac{33}{40}
Answer: The probability is 3340\frac{33}{40}, or 0.825.
Check: Using the overlap rule gives 1840+2140−640=3340\frac{18}{40}+\frac{21}{40}-\frac{6}{40}=\frac{33}{40}. The result is between 0 and 1.

Worked example

Finding a neither probability

At a community event, 80 people are surveyed. Of these, 35 ride a bicycle to the event, 28 take a bus, and 12 do both. What is the probability that a randomly selected person does not ride a bicycle and does not take a bus?
Travel to the event
Travel to the eventSurvey counts in each regionBicycleBus23121629Neither

Survey counts in each region

  1. Find the overlap total
    Let A mean riding a bicycle and B mean taking a bus. These events overlap because 12 people do both. The number who do at least one is found by adding the two event counts and subtracting the shared group once.
    35+28−12=5135+28-12=51
  2. Find the number in neither event
    The 80 people are split between those who do at least one and those who do neither. Subtract the at-least-one count from the total.
    80−51=2980-51=29
  3. Calculate the probability
    There are 29 people in the neither group out of 80 surveyed. The requested probability is the count in that group divided by the total.
    P(neither)=2980P(\text{neither})=\frac{29}{80}
Answer: The probability is 2980\frac{29}{80}, or 0.3625.
Check: The Venn regions contain 23 bicycle-only people, 16 bus-only people, 12 in both, and 29 in neither. These add to 80.

Common mistakes and how to avoid them

Adding the probabilities of overlapping events without adjusting for the shared outcomes.
Correction: Subtract the probability of the intersection once, because those outcomes were counted twice.
Treating “A or B” as meaning exactly one event must happen.
Correction: In probability problems, “A or B” normally includes outcomes where both events happen. Follow a different meaning only if the question says “exactly one.”
Subtracting the shared outcomes twice.
Correction: Add the two event counts or probabilities, then subtract the shared part once.
Dividing by the number in one event instead of the total sample space.
Correction: For a random selection from a group, divide the favourable count by the full group size.

Lesson summary

  • Mutually exclusive events cannot happen together, so their probabilities can be added directly.
  • Overlapping events share outcomes. Subtract the shared part once when finding the probability of A or B.
  • A Venn diagram helps organize the two non-overlapping regions, the intersection, and the neither region.
  • For equally likely outcomes, convert a favourable count to probability by dividing by the total.

Check your understanding

Question 1

A fair die is rolled once. What is the probability of rolling a 1 or a 6?
  1. 16\frac{1}{6}
  2. 13\frac{1}{3}
  3. 12\frac{1}{2}
  4. correctIndex
Show answer and explanation
13\frac{1}{3}
The events are mutually exclusive. Two of the six equally likely results are favourable, so the probability is 26=13\frac{2}{6}=\frac{1}{3}.

Question 2

A group has 30 people. Ten like tea, 14 like coffee, and 4 like both. What is the probability that a randomly selected person likes tea or coffee?
  1. 2030\frac{20}{30}
  2. 2430\frac{24}{30}
  3. 1430\frac{14}{30}
  4. correctIndex
Show answer and explanation
2030\frac{20}{30}
The shared group is counted in both event totals, so subtract it once: 10+14−4=2010+14-4=20. The probability is 2030=23\frac{20}{30}=\frac{2}{3}.

Question 3

In a group of 50 people, 19 play soccer, 22 play basketball, and 7 play both. How many play neither sport?
  1. 9
  2. 16
  3. 23
  4. correctIndex
Show answer and explanation
9
The number who play at least one sport is 19+22−7=3419+22-7=34. Therefore, 50−34=1650-34=16 play neither. The correct option is 16.

Key terms

Event
A group of outcomes that match a stated condition.
Mutually exclusive
Describes two events that cannot happen at the same time.
Overlap
Outcomes that belong to both events.
Intersection
The shared outcomes of two events.
Union
The outcomes in event A, event B, or both.
Venn diagram
A diagram that uses overlapping circles to show how groups of outcomes relate.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation P2. It is a study resource, not an official curriculum publication.

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