Learning intention
We are learning to work out how many times we expect an outcome, and to compare that with what actually happened.
Success criteria
- I can calculate an expected frequency from a probability and a number of trials.
- I can record what actually happened accurately.
- I can find the difference between expected and observed.
- I can explain why the two are almost never equal, and why that is normal.
Curriculum
AC9M6P02 — conduct repeated chance experiments and run simulations with an increasing number of trials using digital tools; compare observations with expected results and discuss the effect on variation of increasing the number of trials.
It does not explain why more trials help. That is Lesson 5. Resist finishing the story today — the pooling activity in the Summarise points at Lesson 5 without resolving it.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 7 min | Two-coin standing game — last one standing wins |
| Launch | 10 min | The bag of four. Introduce expected frequency. |
| Explore | 24 min | Pairs build the bag, calculate, run 40 trials, tabulate |
| Summarise | 10 min | Pool the whole class. Ask the right question about it. |
| Review | 4 min | Exit ticket |
What we are learning today
We are learning to work out how many times we expect an outcome, and to compare that with what actually happened.
I will know I can do it when…
- I can calculate an expected frequency from a probability and a number of trials.
- I can record what actually happened accurately.
- I can find the difference between expected and observed.
- I can explain why the two are almost never equal, and why that is normal.
Set your goal for today
Read the success criteria above. Which one is the biggest stretch for you? Write one thing you are going to focus on. You will come back to it at the end.
Not sure? Explaining WHY expected and observed differ is harder than calculating either of them.
How hard is each question?
Every question in this unit is labelled so you know what kind of thinking it needs. The assessment has questions at all three levels.
SF — simple familiar
You have seen this kind of question before and it is clear what to do. One step, friendly numbers.
CF — complex familiar
You know the method, but there is more than one step, or you have to pick which method to use.
CU — complex unfamiliar
Nobody tells you the method. You work out what to do, and explain why you did it.
Warm-up · 7 minutes
Last one standing
Everyone stands. You are going to toss a coin twice. Before each round, students commit to a prediction using their hands:
- Both hands on head — I predict two heads
- Both hands on hips — I predict two tails
- One hand on head, one on hip — I predict one of each
Toss twice. Anyone who predicted wrong sits down. Play until one student is left, then play again.
Most of the room will not know this at the start. After two or three rounds, ask who is still standing and what they chose. The survivors skew heavily towards "one of each", and students notice.
Do not explain it yet. Ask the question — "is one of these three choices better than the others?" — and take a vote. Then move to the Launch, where the bag has exactly the same structure: 1/4, 1/2, 1/4. Students who spotted the coin pattern will recognise the bag, and that recognition is worth more than being told.
Launch · 10 minutes
The bag of four
Hold up an opaque bag. Show the class the contents, then close it: 2 red, 1 blue, 1 green.
Thirty seconds on whiteboards. Red 2/4 = 0.5 = 50%. Blue and green 1/4 = 0.25 = 25% each. Then the real question:
Take answers. Somebody will say 10. Ask how they know. Then formalise it:
Both methods, on the board
| Method 1 — divide, then multiply | Method 2 — decimal first |
|---|---|
| There are 4 counters, so split the 20 draws into 4 equal parts. 20 ÷ 4 = 5. Red is 2 of the 4 counters, so 5 × 2 = 10. |
Red is 2 out of 4, which is 0.5. 0.5 × 20 = 10. |
Both methods are on the conversion mat. Students use whichever they prefer, and a calculator is fine for the dividing.
Explore · 24 minutes
Predict, then find out
Pairs build their own bag with the same composition: 2 red, 1 blue, 1 green. Then:
- Fill in the probability column for all three colours, three ways.
- Calculate the expected frequency for 40 trials, before drawing anything.
- Run 40 trials. Draw, record, put it back. Every time.
- Fill in observed, then difference. Difference = observed − expected, so it can be negative.
What the table should look like
| Colour | Probability | Expected (40) | Observed | Difference |
|---|---|---|---|---|
| Red | 2/4 = 50% | 20 | 23 | +3 |
| Blue | 1/4 = 25% | 10 | 8 | −2 |
| Green | 1/4 = 25% | 10 | 9 | −1 |
| Total | 100% | 40 | 40 | 0 |
One set of results, invented. The differences add to zero — worth pointing at if a pair notices.
1. Replacement. The counter goes back every time. Pairs that deal out four counters and stop have not run 40 trials.
2. The totals row. Observed must total 40 because they did 40 draws. If it does not, the recording went wrong and the pair should say so rather than adjust it. Recording an inconvenient result honestly is a habit this cycle is trying to build, and it is assessed in Part B.
The differences always sum to zero. Both columns total 40, so the differences cancel. A pair who spots this has found something real. Ask them whether it would still be true for any bag.
Complexity bands on the worksheet
SF Calculate expected frequency for a given probability and trial count.
CF Work backwards — given an expected frequency and a trial count, what was the probability? Then handle a bag where the counts are not obvious.
CU A bag of seven, where no expected frequency is a whole number. What does an expected frequency of 5.71 actually mean?
Differentiation
Support
- 20 trials rather than 40 — the pooled class total still gives the full picture.
- Recording table with the probability column already filled in.
- Convert to percentages first, then find that percentage of the trials.
- Conversion mat on the desk.
Core
- 40 trials, all three colours, full table.
- Expected column completed before drawing.
- Difference calculated with sign.
Extension
- A bag of 7: 1 red, 2 blue, 4 green. Expected frequencies for 40 trials are 5.71, 11.43 and 22.86.
- What does an expected frequency of 5.71 mean, when you cannot draw 0.71 of a counter?
- Check: do those three still add to 40? Why must they?
Part C · Show your thinking
A bag has 8 counters. 3 are red. You draw 40 times. How many reds do you expect? Show your working.
You expect 15 blues out of 60 draws. What is the probability of drawing a blue? Explain how you worked it out.
Look at the DIFFERENCE column in Part A. Add the three numbers together. What do you get? Why must that always happen?
2 1/4. 15 out of 60 is 15/60, which simplifies to 1/4. Working backwards from the expected frequency.
3 Zero. Both the expected column and the observed column total 40, because you did 40 draws, so the differences cancel out exactly. A pair who spots this has found something real — ask whether it would be true for any bag.
Talk it through with a partner
Show your partner your answer to question 3. Your partner picks one of these to ask you:
- Would the differences still add to zero if we did 100 draws?
- Could one difference be bigger than all the others put together?
- What would it mean if your differences did NOT add to zero?
Write down the question you were asked and what you said back.
Choose your path
Pick the one that matches where you are right now. You can change your mind.
A bag has 8 counters. 4 are red. You draw 40 times. How many reds do you expect?
A spinner has 4 equal sections. 1 is shaded. You spin 20 times. How many times do you expect shaded?
Look at the DIFFERENCE column in Part A. Add the three numbers. What do you get?
A different bag holds 7 counters: 1 red, 2 blue, 4 green. You draw 40 times. Work out the expected number for each colour. None of them is a whole number.
What does an expected frequency of 5.71 mean, when you cannot draw 0.71 of a counter?
Add your three expected numbers together. What do you get? Why must that happen?
★2 It is an average, not a count. Over many sets of 40 draws you would average about 5.71 reds per set. In any single set you get a whole number.
★3 Exactly 40. The probabilities total 1, so the expected frequencies must total the number of trials — even though not one of them is a whole number.
Summarise · 10 minutes
Pool the whole class
Each pair adds their red count and their trial count to the class pooling chart. Total both columns.
It almost certainly got bigger. For one pair at 40 trials, the gap between observed and expected reds averages about 2.5. Pool thirteen pairs into 520 trials and the average gap grows to about 9.
What shrinks is the gap as a fraction. One pair is out by about 0.06 of their trials; the pooled class is out by about 0.02. That is the thing that improves, and it is the whole basis of Lesson 5.
So ask this instead: "Write the pooled reds as a fraction of all the trials. How close is that to one half?" Then have one pair write their own result as a fraction of 40 next to it. The class fraction will be visibly closer to 0.5 than almost any individual pair's.
If a student does raise the raw difference getting bigger, that is a good observation, not a wrong one. Say so, and say that Lesson 5 will sort out which number to watch.
Three things to land
- Expected frequency is calculated. Observed frequency is counted. One comes from the maths, the other from the world.
- They are almost never equal, and that is not a mistake. Nobody did anything wrong.
- The bigger the number of trials, the closer the fraction gets to the probability. Point at the pooled chart. Then stop — why this happens is Lesson 5.
Vocabulary
Add to the Frayer boards: trial, frequency, expected, observed. Four words is a lot for one lesson — do expected and observed together as a class and leave the other two for students to complete.
Review · 4 minutes
Exit ticket
How many times do you expect it to land on shaded? Show your working.
Sam spun it 30 times and it landed on shaded 9 times. Did Sam make a mistake? Explain.
Sam says "the probability of shaded is 9 out of 30". Is Sam right? Explain your answer.
1. 12. Method 1: 30 ÷ 5 = 6, then 6 × 2 = 12. Method 2: 2 ÷ 5 = 0.4, then 0.4 × 30 = 12. Accept either, but the working must be there — an unsupported 12 does not show which method the student holds.
2. No. 9 is not 12, but observed results are almost never exactly equal to expected. Nothing went wrong.
3. No. 9 out of 30 is what Sam observed. The probability is 2/5, which comes from the spinner itself and does not change because of what happened when Sam spun it.
Question 3 is the discriminator, and it matters more than it looks. A student who agrees with Sam has swapped observed frequency for probability. That confusion, left alone, makes Lesson 6 impossible — the whole "is this die fair?" task depends on knowing that a result is evidence about a probability rather than the probability itself. Anyone who agrees with Sam should be seated where you can reach them in Lesson 5.
Think back
Your goal
Go back to the Intentions tab and read the goal you set. Did you meet it?
Nothing you type here is saved. Copy it into your book before you close the page.
What you need
- resources/l4-01-worksheet-core.pdf — 3 pages, most of the class
- resources/l4-02-worksheet-support.pdf — 3 pages, 20 trials and a pre-filled probability column
- resources/l4-03-class-pooling-chart.pdf — one per class, A3 if you can. Used in the Summarise and again in Lesson 5.
- shared/conversion-mat.pdf — both methods are on it
- shared/frayer-boards.pdf — print trial, frequency, expected, observed
Equipment
- Counters: 2 red, 1 blue, 1 green per pair. Add a 7-counter set for extension pairs.
- Opaque bags or envelopes — students must not be able to see in
- Coins for the warm-up, one per table group
- The Lesson 2 butcher's paper, brought back to the front
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document and are not repeated here.
- Trial count reduced to 20 where 40 is not sustainable. The pooled class chart still supplies the large-sample picture, so nothing is lost from the Summarise.
- Recording table pre-formatted with the probability column already completed, so the cognitive load sits on the expected-frequency calculation rather than on yesterday's conversions.
- Short trial blocks. Ten draws, mark the sheet, ten more. Forty consecutive draws is a long time to sustain attention on a repetitive task.
- Named roles before drawing starts — one student draws and replaces, the other records, then swap at the halfway mark.
- The warm-up is a standing game. Students who prefer not to stand can play seated with hands on the desk; the prediction signal works the same way.
Misconceptions to expect today
"The observed result IS the probability"
A pair draws 23 reds out of 40 and writes the probability as 23/40. This is the most important misconception in the whole cycle and today is where it becomes visible.
Response. "What was the probability before you started drawing? Did the counters change?" The bag did not change, so the probability did not change.
"I got 23 instead of 20, so I did it wrong"
Students treat the gap as an error and want to re-run until they get exactly 20. Some will quietly adjust their tally.
Response. Ask the room for the biggest gap anyone got, and celebrate it. Then ask whether anyone got exactly the expected number — usually one or two pairs, out of luck rather than skill.
"Expected means it will happen"
The everyday meaning of "expect" overrides the mathematical one. Students predict that the result will be exactly the expected frequency and are surprised every time.
Response. The everyday-versus-mathematical chart. "I expect my friend at four o'clock" does not mean they arrive at four o'clock exactly.
Drawing without replacement
Carried over from Lesson 3 and still the most common procedural error. With only four counters it wrecks the results fast.
Response. Catch it in the first two minutes of the Explore. After that, the pair has 40 trials of unusable data.
What comes next
Lesson 5 — Simulation at scale. The Chance Simulator arrives, and with it the ability to run 2000 trials instead of 40. Lesson 5 answers the question this lesson raises: what exactly is it that gets better when you run more trials?
Before Lesson 5: test the Chance Simulator on the actual devices. Item A5 of the assessment depends on it, and finding out it is blocked by the filter during a lesson is an expensive way to learn that.