'A and B' means both happen; 'A or B' means at least one happens (both included); 'not A' is everything else. Read them from a Venn diagram, a two-way table or a grid, counting each outcome once.
A raffle prize goes to someone who plays sport or music. In a class of 30, 20 play sport and 13 play music. How many are in the draw?
Probability questions often join events with and, or and not. On a Venn diagram, each one is a set of regions.
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.
Sport or music
TryTry it on the board and compare the result with your prediction.
Choose the regions
Words
Regions
In the Sport and Music class
A and B
the overlap only
8
A or B
anything inside the circles (one, the other, or both)
12 + 8 + 5 = 25
not A
everything outside A
5 + 5 = 10
neither
outside both circles
5
In maths, or includes both. In everyday talk "or" sometimes means one but not both ("tea or coffee?"); in maths that is exactly one. To find P(A or B) from totals, add them and take away the overlap once: 20 + 13 − 8 = 25.
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Choose the students who play sport or music. How many are there? Compare with your prediction.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Counting once
Here is the routine for "or" from totals.
Worked exampleBefore each step, say what you think comes next.
35+28−15=48Add the totals, then take the overlap away once.
P(juice or milk)=6048Everyone inside the circles.
P(neither)=1−6048=6012Neither is the complement of "or".
Read Zoe's working. Where are the 4 students with both?
30 students: 13 have a dog, 10 have a cat and 4 have both. What is P(a dog and a cat)?
Zoe works out P(dog or cat) for the same class. Click the line where she goes wrong.
ModifyChange the problem on purpose and see what happens.
Not, or and neither
Need a hint?
Not music is everything outside the Music circle.
Need a hint?
A sum over 9 is 10, 11 or 12.
Another hint
Doubles run down the diagonal. If you chose the sums over 9 first, two doubles are already chosen.
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Choose the students who do not play music.
Use the board. This ticks when you get there.
Two dice: choose every outcome with a sum over 9 or a double.
Use the board. This ticks when you get there.
30 students: 13 have a dog, 10 have a cat and 4 have both. What is P(neither)?
PractiseA few on your own, to check you have it.
And, or, not
A two-way table of 50 people: 12 wear glasses and are left-handed, 18 wear glasses and are right-handed, 5 are left-handed without glasses, 15 neither. What is P(glasses or left-handed)?
For the same 50 people, what is P(left-handed and no glasses)?
P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?
In the Sport and Music class (12 sport only, 8 both, 5 music only, 5 neither), what is P(exactly one of sport and music)?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
An 'or' event
Write your events A and B in words, and how many outcomes are in A, in B and in both.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Pick two events that overlap, such as 'the second dice is 1' and 'the sum is 5 or less'.
Another hint
Count each, then take away the outcomes in both.
Two dice: make an event of the form 'A or B', where A and B overlap, with exactly 15 outcomes.
Use the board. This ticks when you get there.
For teachers
Students find P(A and B), P(A or B), P(not A) and P(neither) from Venn diagrams, two-way tables and two-dice grids, counting the overlap once (AC9M8P02, node 2). About 40 minutes.
I can find 'and', 'or', 'not' and 'neither' on a Venn diagram, a table or a grid.
I can work out P(A or B) without counting the overlap twice.
I can explain that 'or' in maths includes both.
Before class
Do the Run board yourself first. Students need Venn diagrams and two-way tables, grids, and complements (the last three lessons).
The one idea: 'and' is the overlap, 'or' is everything inside the circles (one, the other or both), 'not' is outside a circle, and 'neither' is outside both. The common mistake is adding the totals for 'or' (Zoe's mistake), counting the overlap twice.
Inclusive 'or' is the maths meaning; everyday 'or' is often exclusive ("tea or coffee?"). Name the difference once. The addition rule P(A or B) = P(A) + P(B) − P(A and B) appears as a routine, not a formula to memorise.
Wider: Venn diagrams with three circles, and checking 'or' with a complement (neither).
Running the lesson
Predict: ask whether a probability can be more than 1.
Run: ask how many times the 8 are counted now.
Investigate: ask Zoe where the 4 with both are.
Modify: ask which outcomes are both over 9 and doubles.
Make: ask how many outcomes are in A, in B and in both.
The hard part (on purpose)
The overlap in 'or'. Ask students to point to the overlap and say how many times they have counted it.
When students get stuck
A student adds the two totals for 'or'. Ask: Is anyone in both totals? Fix: Add, then take away the overlap once.
A student gives 'and' as everyone in either circle. Ask: Does 'and' need both to be true? Fix: And is the overlap only.
A student leaves out the 'both' people from 'or'. Ask: Does someone with a dog and a cat have a dog or a cat? Fix: In maths, 'or' includes both.
Stretch and support
Stretch: Find P(exactly one of A and B) for the Sport and Music class, and write it using 'or', 'and' and 'not'.
Support: Four mini Venn diagrams with 'and', 'or', 'not A' and 'neither' shaded, as a reference card.
Where it leads
This completes AC9M8P02.
Where it leads: simulations of compound events (next lesson), and independent events in Year 10.
What each step is for
sport or music: Committing to P(sport or music) shows whether they add the totals and count the overlap twice.
choose the regions: Choosing 'or' on the diagram shows it is three regions, with the overlap counted once.
counting once: 'And' is one cell; 'or' adds the totals then takes the overlap once; adding the totals alone counts both twice (Zoe).
not, or and neither: 'Not' and 'or' on the diagram and on a two-dice grid, and 'neither' as the complement of 'or'.
and, or, not: Reading and, or, not and exactly one from tables and totals, mixed; counting the overlap twice is the planted slip; exactly one separates the maths or from the everyday one.
an 'or' event: Building an 'or' event to a given size makes them handle the overlap on purpose.
Now you can
Find 'and', 'or', 'not' and 'neither' on a Venn diagram, a table or a grid.
Work out P(A or B) without counting the overlap twice.