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Compound events, the addition rule and mutual exclusivity

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Compound events

A compound event combines two or more events. Translate the words into regions before adding probabilities, and subtract a shared region once when it has been included twice

\[\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)\]
  1. 1Use intersection for both An intersection contains outcomes in both named events \(\displaystyle A\cap B\)
  2. 2Use union for at least one A union contains outcomes in either event, including the overlap \(\displaystyle A\cup B\)
  3. 3Use only-A for A but not B Remove the B part from event A \(\displaystyle A\cap B′\)
  4. 4Subtract overlap once Adding two marginal probabilities counts the shared region twice \(\displaystyle \frac{1}{2}+\frac{2}{5}-\frac{1}{10}=\frac{4}{5}\)
Year 11, VCE Units 1 & 2, Chapter 8, 4 difficulty levels, No calculatorBuild a worksheet

Compound events

A compound event combines two or more events. Translate the words into regions before adding probabilities, and subtract a shared region once when it has been included twice

\[\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)\]
  1. Use intersection for both An intersection contains outcomes in both named events \(\displaystyle A\cap B\)
  2. Use union for at least one A union contains outcomes in either event, including the overlap \(\displaystyle A\cup B\)
  3. Use only-A for A but not B Remove the B part from event A \(\displaystyle A\cap B′\)
  4. Subtract overlap once Adding two marginal probabilities counts the shared region twice \(\displaystyle \frac{1}{2}+\frac{2}{5}-\frac{1}{10}=\frac{4}{5}\)

Worked example

A synthetic classification records two properties for 30 cases. Event A contains 18 cases, event B contains 15 cases, and 5 cases are in both events. A case is chosen uniformly at random. Find the probability that it is in neither event

\[\Pr(A′\cap B′)\]
  1. \(\displaystyle n(A\cup B)=18+15-5=28\)The union count subtracts the 5 cases in both events once
  2. \(\displaystyle n(A′\cap B′)=30-28=2\)The cases outside the union are in neither event
  3. \(\displaystyle \Pr(A′\cap B′)=\frac{2}{30}=\frac{1}{15}\)Divide the neither count by 30 and reduce the fraction

Common mistakes

Counting the overlap twice
✕\(\displaystyle \begin{aligned}\Pr(A)&=\frac{1}{2}\\\Pr(B)&=\frac{2}{5}\\\Pr(A\cap B)&=\frac{1}{10}\\\Pr(A\cup B)&=\frac{1}{2}+\frac{2}{5}=\frac{9}{10}\end{aligned}\)✓\(\displaystyle \begin{aligned}\Pr(A)&=\frac{1}{2}\\\Pr(B)&=\frac{2}{5}\\\Pr(A\cap B)&=\frac{1}{10}\\\Pr(A\cup B)&=\frac{1}{2}+\frac{2}{5}-\frac{1}{10}=\frac{4}{5}\end{aligned}\)
The overlap is included in both marginal probabilities, so it must be subtracted once
Swapping union and intersection
✕\(\displaystyle A\cup B=A\cap B\)✓\(\displaystyle A\cap B \text{ means both, whereas } A\cup B \text{ means at least one}\)
Intersection means both events occur, while union includes either event and their overlap
Taking a complement without using one
✕\(\displaystyle \Pr(A′\cap B′)=\Pr(A\cup B)\)✓\(\displaystyle \Pr(A′\cap B′)=1-\Pr(A\cup B)\)
The probability outside a union is its complement, found by subtracting the union probability from 1

Difficulty levels

introRecognise union, intersection and complement language and add disjoint event regions
standardUse the addition rule when two events have a nonzero overlap
challengeRecover an intersection or combine exactly-one and neither regions
examIdentify the event relationship and calculate one exact compound-event probability

Questions about this topic

How do union and intersection differ?+

A union contains outcomes in at least one of the named events, including their overlap; an intersection contains outcomes shared by both events

\[\begin{aligned}\text{Union: }&A\cup B\\\text{Intersection: }&A\cap B\end{aligned}\]
Why is the intersection subtracted in the addition rule?+

The intersection is counted once in each marginal probability, so adding the marginals counts it twice. Subtracting it once leaves one count

What does mutually exclusive mean?+

Mutually exclusive events cannot occur together, so their intersection is empty and has probability zero. This is different from independence

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 8: Probability and counting

In this chapter

Finite events, probability assignments and complementsRepresent a multi-stage finite sample spaceCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConditional probability from exact representationsDistribution of a finite random variableDesign and interpret a probability simulationCount restricted selections and calculate exact probabilitiesCount arrangements with product, sum and restriction rulesTest exact independenceExact probability of repeated independent successesMulti-stage probability, multiplication and total probabilityProbability from a uniform area modelProbabilities of sequences of independent events