Studio Aletheia · The Center for Quantitative Studies
6.DPSR.2.3SC 6th Grade Math
Complements in Two Directions
Lesson 22 · Guided Practice, Calculate the Complement of an Event
Yesterday you found a complement when you already knew the event's probability. Today you'll work in both directions, combining the complement rule with the probability calculations from earlier lessons.
Learning Targets and Success Criteria
Today you'll practice moving between outcome counts, probabilities, and complements in any order.
What I will learn
- I can calculate an event's probability from outcome counts, then find its complement.
- I can work backward from a complement's probability to find the original event's probability.
- I can explain why the complement rule works using outcome counts as proof.
- I can solve multi-step problems that combine probability and complement calculations.
What success looks like
- I can calculate at least three complements starting only from outcome counts.
- I can find an original event's probability when only given its complement's probability.
- I can explain the complement rule using a specific outcome-count example.
- I can solve a two-step problem involving both a probability calculation and a complement calculation.
Build fluency calculating complements from outcomes and from given probabilities.
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.
Two Ways to the Same Answer
Actuaries constantly check their work by calculating a probability two different ways and confirming both answers match, exactly like you'll practice today. That habit of double-checking with the complement rule is part of what makes their pricing numbers trustworthy.
Working the Rule Both Ways
Today you'll practice moving between outcome counts, probabilities, and complements in any order.
So far you've mostly started with a given probability and found its complementary eventThe event that everything that is NOT the original event happens instead; P(not A) = 1 − P(A).. Today you'll also start from raw outcome counts. If a bag has 25 counters and 5 are red, you first calculate P(red) = 5/25 = 1/5, and only then apply the complement rule to get P(not red) = 1 − 1/5 = 4/5. Two steps, but each one is a tool you already have.
You can also run the rule in reverse. If you're told that the probability of NOT drawing a winning ticket is 9/10, you can find the probability of drawing a winning ticket by subtracting the other way: P(winning) = 1 − 9/10 = 1/10. The complement rule always connects two probabilities that add to exactly 1, so whichever one you're missing, subtraction from 1 gets you there. Mixing this with probability calculations from outcome counts is what today's harder problems will ask you to do.
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 bag of 25 or 50 marbles
- B. A spinner divided into equal sections
- C. Fraction/decimal/percent conversion chart
- D. Your Data Journal
- E. A calculator (optional)
When a problem gives outcome counts, calculate the event's probability first, and only then apply the complement rule.
Complement Lab: Round Two
Today's lab mixes outcome-count problems with reverse complement problems.
Two-Step Complement Challenge
You'll solve all four scenarios, showing both the probability step and the complement step.
Data Journal Entry
Accountability Checklist
Find the Missing Piece
A bag has 100 marbles, and 15 are purple. Calculate P(purple), then use it to find the probability of the complementary event. Show both steps.
The Center for Quantitative Studies · Course Navigation
Move between lessons or return to the larger Studio Aletheia ecosystem.