AP®StatisticsCollege BoardStudy GuidesUnit 4: Probability, Random Variables & Probability DistributionsProbabilityIntroduction to Combined Events

Introduction to Combined Events (College Board AP® Statistics)

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Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Intersections & unions

What is the intersection of events?

  • The intersection of two events A and B is the set of outcomes that are included in both A and B

  • The intersection of events A and B is denoted A intersection B

    • A intersection B is identical to B intersection A

Table with Bronze, Silver, and Gold rows, and columns numbered 1 to 6. The columns for the even numbers are highlighted and the row for gold is highlighted. The outcomes cells that are in both a highlighted row and column form the intersection of the events of getting gold and rolling an even number.
Example of the intersection of two events

  • The intersection of any number of events is the set of outcomes that are included in all of the events

What is the union of events?

  • The union of two events Aand B is the set of outcomes that are included in A or B or both

  • The union of events A and B is denoted A union B

    • A union B is identical to B union A

Table with Bronze, Silver, and Gold rows, and columns numbered 1 to 6. The columns for the even numbers are highlighted and the row for gold is highlighted. The outcomes in the highlighted cells form the union of the events of getting gold and rolling an even number.
Example of the union of two events

  • The union of any number of events is the set of outcomes that are included in at least one of the events

  • For any two events, A and B, the following events contain the same outcomes:

    • the intersection of the complements of the events, A apostrophe intersection B apostrophe

      • the outcomes that are not in A and not in B

    • the complement of the union of the events, open parentheses A union B close parentheses apostrophe

      • the outcomes that are not in either A or B or both

  • Therefore A apostrophe intersection B apostrophe and A union B are complementary events

A table showing outcomes with bronze (B), silver (S), and gold (G) across numbers 1 to 6. Highlighted cells indicate combinations involving gold or even numbers. The cells that are not highlighted are the ones containing odd numbers and bronze or silver.
Example of the relationship between the union of two events and the intersection of the complements of the two event

Examiner Tips and Tricks

The fact that A apostrophe intersection B apostrophe and A union B are complementary events can be used to find probabilities.

If you are asked to find the probability that A or B occurs, you can find the probability that neither A nor B occurs and subtract that from 1.

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.