The sample space lists every possible outcome. When the outcomes are equally likely, the probability of an event is favourable outcomes ÷ total outcomes; when they are not (a spinner with a bigger sector), use the sizes. The probabilities add to 1.
A spinner is half red. Red, blue or green: three colours, so is each one a third? Spin it a thousand times and see.
Probability measures how likely something is, from 0 (impossible) to 1 (certain).
The sample space is the list of every possible outcome: a coin {H,T}, a dice {1,2,3,4,5,6}.
An event is one or more of those outcomes, such as "an even number".
When the outcomes are equally likely, P(event)=total outcomesfavourable outcomes.
When they are not (a spinner with a bigger sector, a bag with more red counters), use the sizes or counts. Either way, the probabilities of all the outcomes add to 1.
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.
A third each?
Three colours, but are they the same size?
TryTry it on the board and compare the result with your prediction.
Spin it
The table lists the sample space, each outcome's probability, and what happened.
Try it yourselfYour edits are saved in this browser.
Run at least 1000 spins. Is the relative frequency of blue nearer 41 or 31? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
List, then count
Here is how to find a probability.
Worked exampleBefore each step, say what you think comes next.
{1,2,3,4,5,6}List the sample space: every possible outcome of rolling a dice.
equally likely?✓A fair dice: each face has the same chance.
even={2,4,6}The favourable outcomes for the event.
P(even)=63=21=0.5=50%Favourable ÷ total, in any form.
P(red)=21,P(blue)=P(green)=41Not equally likely: use the sizes. Check: they add to 1.
Now work out P(even) on a dice.
Read Kai's reasoning. Is each colour the same size?
Roll a fair dice. What is the probability of an even number? Give a fraction in simplest form.
Kai works out probabilities for the half-red spinner. Click the line where his reasoning goes wrong.
A bag has 3 red, 5 blue and 2 green counters. You take one without looking. What is P(blue)? Simplest form.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Roll a fair dice. What is the probability of a number greater than 4? Simplest form.
A spinner has P(red)=0.5 and P(blue)=0.3. The only other colour is green. What is P(green)?
Write P(heads) for a fair coin as a percentage.
PractiseA few on your own, to check you have it.
Probabilities
A bag has 4 yellow and 6 purple counters. P(yellow)? Simplest form.
Pick a letter at random from the word MATHS. What is the probability it is a vowel? (A fraction.)
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Design a bag
Then: design a bag with P(red)=31. What is the smallest bag that works?
Design a bag of 10 counters with P(red)=103 and P(blue)=21; the rest are green. How many green counters?
For teachers
Students list sample spaces for coins, dice, bags and spinners, assign probabilities as favourable ÷ total when outcomes are equally likely and from sizes or counts when they are not, check that probabilities add to 1, and test a prediction with 1000 spins (AC9M7P01, nodes 1 and 2). About 45 minutes.
I can work out a probability as a fraction, decimal or percentage.
I can tell when outcomes are not equally likely, and use sizes or counts instead.
Before class
Do the Run board first. Students need fractions, decimals and percentages (Number Foundations).
This is a hinge: everything in Chance builds on a complete sample space and correct probabilities. The common mistake is treating every listed outcome as equally likely (three colours, so 1/3 each); it is planted in the Predict and in Kai's reasoning.
The board's random spins are different every time; with 1000 or more spins the relative frequencies are reliably close to the probabilities.
Running the lesson
Predict: take a show of hands; ask students who chose 1/3 how they decided.
Run: ask what relative frequency you would expect if each colour really were 1/3.
Investigate: write the sample space for each device on the board before any probability.
Modify: for repeated labels, draw the spinner with A, A, B.
Make: ask for the smallest bag with P(red) = 1/3.
When students get stuck
A student gives each colour 1/3 (Kai's mistake). Ask: Is each colour the same size? Fix: Use the sizes: half the spinner is 1/2.
A student counts colours instead of counters. Ask: How many counters are there in total? Fix: Favourable counters ÷ total counters.
A student gives the count instead of the probability (3 instead of 1/2). Ask: Out of how many? Fix: Divide by the total number of outcomes.
Stretch and support
Stretch: Design a spinner with four colours where one colour is three times as likely as each of the others.
Support: List the sample space in a table with a probability column; check the column adds to 1.
Where it leads
Next: Predict and test (AC9M7P01 node 3, AC9M7P02 node 1): expected counts and experiments.
Where it leads: complementary events and two-step experiments (Year 8).
What each step is for
a third each?: Committing to P(blue) on a half-red spinner shows whether they assume the listed outcomes are equally likely.
spin it: Spinning a thousand times shows blue settling near a quarter, not a third: the outcomes in a sample space need not be equally likely.
list, then count: List the sample space completely, check whether the outcomes are equally likely, then favourable ÷ total, or the sizes; the probabilities add to 1. Kai's mistake treats every listed outcome as equally likely.
one change at a time: One change at a time: a different event, a spinner with repeated labels, a missing probability, a percentage, then the edge cases of impossible and certain.
probabilities: Sample spaces and probabilities for coins, dice, bags and spinners, mixed; the equally-likely slip planted again; the review keeps comparing data fresh.
design a bag: Designing a bag to match given probabilities reverses the lesson: from probabilities back to counts, using equivalent fractions.
Now you can
List the sample space for a single event.
Work out a probability as a fraction, decimal or percentage.
Tell when outcomes are not equally likely, and use sizes or counts instead.