Studio Aletheia · The Center for Quantitative Studies
6.DPSR.2.3SC 6th Grade Math
Calculate the Complement of an Event
Lesson 21 · New Content, Calculate the Complement of an Event
Every event has an opposite: everything that happens when the event doesn't. Today you'll learn a shortcut for finding the probability of that opposite, called the complement, using nothing more than subtraction.
Learning Targets and Success Criteria
Today you'll learn a shortcut for finding the probability of everything an event is not.
What I will learn
- I can identify the complement of a given event.
- I can calculate the probability of a complement using P(not A) = 1 − P(A).
- I can explain why an event and its complement always add up to 1.
- I can check a complement calculation by adding the two probabilities together.
What success looks like
- I can state the complement of at least three different events correctly.
- I can calculate at least three complement probabilities using subtraction.
- I can explain in a sentence why P(A) + P(not A) always equals 1.
- I can verify my complement calculation by checking that both probabilities sum to 1.
Calculate the probability of everything that is not the event.
The words we'll use in today's lesson.
These terms will carry through today's mini-lesson, Data Lab, hands-on activity, journal, and exit challenge, and they'll keep coming back all year.
Complementary Event
The event that everything that is NOT the original event happens instead; P(not A) = 1 − P(A).
Where this shows up in the real world.
Thinking like a mathematician is not just a school skill. It's what people get paid to do every day, in jobs you may not have heard of yet.
The Odds of Nothing Happening
Insurance only makes sense because most years, nothing bad happens — and actuaries need to know that probability just as precisely as the probability of a claim. Soon you'll see that the shortcut you're learning today, P(not A) = 1 − P(A), is exactly how they calculate it.
Everything Else That Could Happen
Today you'll learn a shortcut for finding the probability of everything an event is not.
Take any event, like rolling a 3 on a number cube. There's exactly one way for it to happen, and five ways for it not to happen (rolling a 1, 2, 4, 5, or 6). That second group, everything that is NOT the original event, is called the complementary eventThe event that everything that is NOT the original event happens instead; P(not A) = 1 − P(A).. Together, an event and its complement always cover every single possible outcome — there's nothing left over.
Because an event and its complement cover everything, their probabilities always add up to exactly 1. That gives you a shortcut: instead of counting all the outcomes in the complement, you can just calculate P(not A) = 1 − P(A). If the probability of rolling a 3 is 1/6, the probability of not rolling a 3 is 1 − 1/6, which is 5/6, and you never had to count those five outcomes one at a time. This shortcut works no matter how big or complicated the total outcomes get, which is exactly why it's worth learning well.
Adapted from Studio Aletheia's The Center for Quantitative Studies curriculum library, drawing on mathematical resources and the SC CCR Mathematics Standards.
Materials for the Data Lab.
- A. A number cube (1-6)
- B. A spinner divided into equal sections
- C. A bag of 10 or 20 marbles
- D. Your Data Journal
- E. A calculator (optional)
Whenever you're asked for 'not' an event, subtract the event's probability from 1 instead of recounting outcomes.
Complement Lab
Today's lab gives you a probability for an event, and you'll use the complement rule to find the rest.
Find What's Left
You'll calculate complements for all four scenarios and verify each one adds up to 1.
Data Journal Entry
Accountability Checklist
What's Not Happening?
A spinner has 8 equal sections, and the probability of landing on green is 3/8. Find the probability of the complementary event and explain what it represents.
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