Probability · Lesson 2 of 8
Combining and Classifying Events
“Use set operations to describe combined conditions and distinguish overlap from complete coverage.”
• Translate not, or, and, and but not into event-set operations. • Construct unions, intersections, differences, and complements relative to S. • Test whether events are mutually exclusive or exhaustive. • Recognise a mutually exclusive and exhaustive family as a partition of S. • Justify event relationships using sample points rather than verbal intuition.
Changing the question changes the set
Two events may describe different features of the same result. For one die, A could mean prime and B could mean odd. We can ask whether either condition holds, whether both hold, or whether one holds without the other. These combined questions are expressed by the same operations used to combine sets.
Always keep the sample space fixed while combining events. The words not A refer to outcomes of that experiment that are outside A; they do not include anything outside S. For a six-sided die, 8 is not a prime face that failed to appear. It is not a possible face in this model at all.
The event A′=S−A containing every outcome in S that does not belong to A. It is read “not A”.
Problem
Let A mean exactly one tail in three tosses. Find not A.
- 1.Exactly one tail gives A={HHT,HTH,THH}.
- 2.Start with S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} and remove the three members of A.
- 3.A′={HHH,HTT,THT,TTH,TTT}. It includes zero tails, two tails, and three tails: every case that is not exactly one tail.
Or, and, and but not
The word or in event algebra is inclusive: A or B permits A alone, B alone, and both together. The word and keeps only the common outcomes. But not removes the outcomes satisfying the second condition from those satisfying the first.
A∪B, the event that at least one of A and B occurs, including the possibility that both occur.
A∩B, the event that A and B both occur.
A−B, the event that A occurs and B does not occur.
| Description | Set | Membership test |
|---|---|---|
| A or B | A∪B | In A, in B, or in both |
| A and B | A∩B | In both sets |
| A but not B | A−B=A∩B′ | In A and outside B |
| Not A | A′ | In S and outside A |
The centre is not a fourth event to be added to an already complete union; it is the region shared by A and B. This distinction will later explain why adding their separate probabilities can count the shared part twice. For now, look at which outcomes belong to each region.
Problem
For a die, let A={2,3,5} be prime and B={1,3,5} be odd. Find A or B, A and B, A but not B, and not A.
- 1.The union collects all distinct members: A∪B={1,2,3,5}. A repeated member is written only once.
- 2.The intersection keeps the common values: A∩B={3,5}.
- 3.Removing 1,3,5 from A leaves A−B={2}. Equivalently, intersect A with B′={2,4,6}.
- 4.The complement is found from S={1,2,3,4,5,6}: A′={1,4,6}. Note that 1 is not prime.
Problem
A die is thrown twice. Let A mean that the first score is 6, and B that the total is at least 11. Find A∩B.
- 1.Use ordered pairs (first score,second score). A={(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}.
- 2.The totals at least 11 occur at B={(5,6),(6,5),(6,6)}.
- 3.The common outcomes are (6,5) and (6,6), so A∩B={(6,5),(6,6)}.
- 4.The result means “first score 6 and total at least 11”. The outcome (5,6) is excluded because its first score is not 6.
Inclusive “or” allows both events. “Exactly one of A and B” excludes the intersection and instead uses (A−B)∪(B−A). The difference A−B is also generally different from B−A.
When two events cannot occur together
If two events have no shared outcome, observing one makes the other impossible on that same performance of the experiment. This relationship concerns overlap, not which event is more likely. We can decide it by testing their intersection.
Events A and B are mutually exclusive when A∩B=∅. No outcome can make them both occur.
On one die, odd and even are mutually exclusive because {1,3,5} and {2,4,6} share no value. Odd and less than 4 are not mutually exclusive because 1 and 3 belong to both. A common point is enough to disprove mutual exclusivity; you do not need every point to be common.
Distinct simple events are always mutually exclusive. For example, {2} and {5} cannot occur on the same single roll. With several events, pairwise mutually exclusive means that every distinct pair has empty intersection. Checking only one pair is insufficient.
Problem
Let A mean an even total, B a total divisible by 3, C a total less than 4, and D a total greater than 11. Which pairs are mutually exclusive?
- 1.The sample space contains 36 ordered pairs. C={(1,1),(1,2),(2,1)} and D={(6,6)}.
- 2.A and B overlap at total 6 and total 12: their common pairs are (1,5),(2,4),(3,3),(4,2),(5,1),(6,6).
- 3.A and C share (1,1); A and D share (6,6). B and C share (1,2) and (2,1); B and D share (6,6). These four pairs of events are not mutually exclusive.
- 4.C and D have no common outcome because a total cannot be both less than 4 and greater than 11. Therefore C,D is the only mutually exclusive pair among the six pairs.
Covering every possible outcome
A different question asks whether a family of events leaves any outcome unaccounted for. Events are exhaustive when their union is S, so at least one member must occur whenever the experiment is performed. This can happen even if some members overlap.
A family E₁,…,Eₙ is exhaustive if E₁∪⋯∪Eₙ=S. Every sample point belongs to at least one event in the family.
For a die, A={1,2,3}, B={3,4}, and C={5,6} are exhaustive because their union contains all six faces. They are not mutually exclusive because A and B share 3. By contrast, the odd and even events are both mutually exclusive and exhaustive: they have no overlap and no gap.
Two events can also be mutually exclusive without being exhaustive. The events {1} and {6} do not overlap, but outcomes 2,3,4,5 belong to neither. These examples separate the two tests: an empty intersection says nothing by itself about complete coverage.
Problem
For three tosses, use A: no head, B: exactly one head, C: at least two heads. Are these events mutually exclusive and exhaustive?
- 1.A={TTT}, B={HTT,THT,TTH}, C={HHT,HTH,THH,HHH}.
- 2.Their sizes are 1,3,4 and their union is the full eight-point sample space, so the family is exhaustive.
- 3.No outcome has both zero and one head, zero and at least two heads, or one and at least two heads. Thus every pairwise intersection is empty.
- 4.The family is both mutually exclusive and exhaustive. Its sets form a partition: exactly one category describes any observed sequence.
A and A′ are always mutually exclusive and exhaustive. They do not overlap, and together they cover S. This set relationship will later give the complement probability rule.
Quiz
Which expression represents A but not B?
A={1,2,3} and B={3,4,5,6} on a die. Which description is correct?
Which outcome proves that prime and odd events on a die are not mutually exclusive?
What does the event A∪B permit?
For two tosses, A={HH} and B={TT}. Which statement holds?
Which property holds for A and its complement in any sample space?
Practice Problems
- A die is rolled. Let E={4} and F={2,4,6}. Decide whether E and F are mutually exclusive and justify your answer with a sample point.
- For one die, write A: less than 7, B: greater than 7, C: a multiple of 3, D: less than 4, E: even and greater than 4, F: not less than 3. Find A∪B, A∩B, B∪C, E∩F, D∩E, A−C, D−E, E∩F′, and F′.
- For two ordered dice let A mean total greater than 8, B mean that at least one die shows 2, and C mean a total at least 7 divisible by 3. Describe each event and determine every mutually exclusive pair.
- For three coins let A mean three heads, B exactly two heads, C three tails, and D a head on the first coin. Classify simple and compound events, and identify every mutually exclusive pair.
- For three tosses, give examples of: two exclusive events; three exclusive and exhaustive events; two overlapping events; two exclusive but nonexhaustive events; and three exclusive but nonexhaustive events. Verify each choice.
- For two dice define A: first die even, B: first die odd, C: total at most 5. Describe A′, B′, A∪B, A∩B, A−C, B∪C, B∩C, and A∩B′∩C′.
- Using the events in the preceding problem, decide with reasons whether: A,B are exclusive; A,B are exclusive and exhaustive; A=B′; A,C are exclusive; A,B′ are exclusive; and A′,B′,C form an exclusive and exhaustive family.
Key Takeaways
• Not, or, and, and but not correspond to complement, union, intersection, and difference. • A−B=A∩B′; inclusive or includes the intersection. • Mutual exclusivity means no shared outcome; exhaustiveness means no uncovered outcome. • A partition is a pairwise disjoint exhaustive family, so exactly one category occurs. • A and A′ always partition S.