An event and its complement (everything else that could happen) cover every outcome once. Use that to find probabilities quickly, especially for 'at least one'.
The forecast says a 30% chance of rain. What is the chance it stays dry? You know straight away, but why does it work?
An event is a set of outcomes, such as "roll a 5 or a 6". Its complement, written not A (or A′), is every other outcome in the sample space.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Not red
TryTry it on the board and compare the result with your prediction.
An event and its complement
Every outcome is in A or in not A, never both and never neither. So:
P(A)+P(not A)=1P(not A)=1−P(A)
Try it yourselfYour edits are saved in this browser.
Click the outcomes for A: 'roll at least 5'. Then press Show not A. Do P(A) and P(not A) add to 1? Does that match your prediction?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Using the complement
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
P(red)=83A bag has 3 red counters out of 8.
P(not red)=1−83Not red is every other counter.
=85Check: 5 counters are not red.
Read Ben's working. Which outcomes are in neither event?
The chance a bus is late is 0.15. What is the chance it is not late?
Ben works with two events on a dice. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
At least one
Need a hint?
No six means the first dice is not 6 and the second is not 6.
Another hint
That is the 5 by 5 block of outcomes from 1 to 5. Clicking a row or column number chooses the whole row or column.
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Two dice are rolled. Choose every outcome with no six at all.
Use the board. This ticks when you get there.
So what is the probability of at least one six when two dice are rolled?
A factory finds that 2% of its light globes are faulty. What percentage are not faulty?
PractiseA few on your own, to check you have it.
Complements
P(a team wins) = 0.45. A draw is not possible. What is P(the team does not win)?
Review: a dice is rolled 120 times. How many sixes do you expect?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A quarter
Then describe your event in words, so someone else could check it.
Try it yourselfYour edits are saved in this browser.
Need a hint?
P(A) = 1/4 means 9 of the 36 outcomes.
Another hint
Try a rule, such as 'the first dice is even and the second is 3 or less'. Clicking a row number chooses a whole row.
Two dice: choose an event A with P(A) = 1/4. Then show not A. What is P(not A)?
Use the board. This ticks when you get there.
For teachers
Students choose events on a dice and two dice, show their complements, and use P(not A) = 1 − P(A), including for 'at least one' (AC9M8P01, nodes 1 and 2). About 40 minutes.
I can find P(not A) from P(A), in fractions, decimals and percentages.
I can use the complement to find the chance of 'at least one'.
Before class
Do the Run board yourself first. Students need sample spaces and probability as a fraction (Year 7 Chance).
The one idea: an event and its complement cover every outcome exactly once, so their probabilities add to 1. The common mistake is calling any two events that can't happen together complementary (Ben's mistake); complementary events must also cover everything. "Mutually exclusive" is the formal word for can't-happen-together; it is used in the teacher notes, not required of students yet.
'At least one' is the main use: one block to count (none) instead of many cases.
Wider: the chance of at least one six in four rolls of a dice, 1 − (5/6)⁴, a famous gambling question (needs multiplying, Year 10).
Running the lesson
Predict: ask whether the other colours matter.
Run: ask whether any outcome is in both, or in neither.
Investigate: ask Ben where a roll of 3 goes.
Modify: ask why counting 'no six' is quicker than 'at least one six'.
Make: ask how many outcomes make a quarter of 36.
The hard part (on purpose)
Telling complementary from 'can't happen together'. Ask: does every outcome land in one of the two?
When students get stuck
A student says a heart and a spade are complementary. Ask: Where do clubs go? Fix: Complementary events cover every outcome.
A student counts 12 outcomes with at least one six. Ask: Which outcome did you count twice? Fix: (6, 6) is in both rows; or count no six (25) and subtract from 36.
A student writes P(not A) = P(A). Ask: Which outcomes are not in A? Fix: Count everything else, or subtract from 1.
Stretch and support
Stretch: Two dice are rolled. Which is more likely: at least one six, or a sum of 7 or more? Use complements.
Support: A sample space printed as a grid, with two colours of highlighter.
Where it leads
This covers AC9M8P01.
Where it leads: two-way tables and Venn diagrams (next), and 'and', 'or' and 'not' in compound events.
What each step is for
not red: Committing to P(not red) from P(red) alone shows whether they see that the other colours don't need to be known.
an event and its complement: Choosing an event on a dice and showing its complement shows the two cover every outcome once, so their probabilities add to 1.
using the complement: Subtract from 1 to find a complement; events that can't happen together are not always complementary (Ben); a complementary pair must cover every outcome.
at least one: 'At least one six' has many cases, but its complement 'no six' is one block: count it and subtract.
complements: Complements in decimals, fractions, percentages and words, mixed; calling events that can't happen together complementary is the planted slip; expected count is the Year 7 review.
a quarter: Building an event with a given probability, then its complement, shows they can reason from P to outcomes.
Now you can
Name the complement of an event.
Find P(not A) from P(A), in fractions, decimals and percentages.
Use the complement to find the chance of 'at least one'.