Many trials

Classroom resources

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Students will be able to

• I can describe how the relative frequency changes as the number of trials grows.
• I can use a large experiment to estimate a probability.
• I can explain why a run of one outcome does not make another one 'due'.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run and Make boards first. Students need Predict and test.

The one idea: over many trials the relative frequency settles close to the probability, because each new trial changes the fraction less, not because results balance out. The common mistake is the gambler's fallacy (a coin is 'due' a head); it is planted in Zara's reasoning and again in Practise.

The mystery bag hides its contents: students estimate the make-up from 1000 draws (with replacement). It has 3 red and 7 yellow counters.

Running the lesson

1. Predict: take a show of hands; many students choose 'equally likely'.
2. Run: ask students to press Start again and compare the first ten flips of different runs.
3. Investigate: show the 7/10 → 8/11 and 501/1000 → 502/1001 calculation on the board.
4. Modify: collect the class's estimates for the mystery bag; reveal 3 red, 7 yellow.
5. Make: collect how many flips each student needed.

When students get stuck

Answers to the checks

1. 10 or 1000?: Which is more likely to give a relative frequency of heads between 0.45 and 0.55: 10 flips, or 1000 flips?
   1000 flips. In 10 flips, getting 3 or 7 heads (0.3 or 0.7) is common. Over 1000 flips the relative frequency is almost always within 0.05 of 0.5.
2. watch it settle: Run 10 flips, then keep going to at least 1000. Watch the line. Was your prediction right?
   
3. Settling, not balancing: After 1000 flips, a coin has shown 482 heads. What is the relative frequency of heads? (A decimal.)
   0.482
   482 ÷ 1000 = 0.482: close to 0.5.
4. Find the mistake: Zara thinks about her next flip. Click the line where her reasoning goes wrong.
   line 3
   Line 3: the coin does not remember. Each flip is still 1/2 heads. Five tails in a row is unusual but not impossible, and it is not 'balanced out'; it just matters less as more flips are added.
5. Why it settles: Why does the relative frequency settle down over many trials?
   Each new trial changes the total by less and less, so the fraction moves less
   One more head changes 7/10 a lot, but 501/1000 hardly at all.
6. A dice: Roll at least 2000 times. Does the relative frequency of sixes settle near $\tfrac{1}{6} \approx 0.167$?
   
   After 2000 rolls the relative frequency is usually within about 0.02 of 0.167.
7. A mystery bag: A mystery bag holds 10 counters, red and yellow. Draw (with replacement) at least 1000 times and estimate how many are red.
   
   The relative frequency of red settles near 0.3, so about 3 of the 10 counters are red.
8. Which to trust: Which experiment gives the more trustworthy estimate of P(red) for a spinner?
   5000 spins with 1520 reds
   More trials give a relative frequency closer to the probability: 1520/5000 = 0.304.
9. Long runs: Edge case: in 10 000 flips of a fair coin, will there be any runs of 5 tails in a row?
   Almost certainly, many: long runs happen in long experiments
   In 10 000 flips, a run of 5 tails is expected hundreds of times; the overall fraction still settles near 1/2.
10. Estimate: A drawing pin landed point-up 312 times in 800 throws. Estimate the probability it lands point-up. (A decimal.)
    0.39
    312 ÷ 800 = 0.39. A pin has no sample space of equal outcomes, so experiments are the way to estimate.
11. Streaks: A roulette wheel landed on red 8 times in a row. What is true about the next spin?
    Red is exactly as likely as before
    Each spin is independent: the probability does not change.
12. Use the estimate: Using your estimate of 0.39, how many point-up landings would you expect in 200 throws?
    78
    200 × 0.39 = 78.
13. Review: expected count: Review: how many heads do you expect in 300 flips of a fair coin?
    150
    300 × 1/2 = 150.
14. how many is enough?: Keep flipping until you have at least 1000 flips and the relative frequency of heads is within 0.01 of 0.5.
    
    It usually takes a few thousand flips. Getting within 0.01 is much harder than within 0.1.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › A dice

Open task: any program that does what the step asks.

one change at a time › A mystery bag

Open task: any program that does what the step asks.

one change at a time › Which to trust

Open task: any program that does what the step asks.

one change at a time › Long runs

Open task: any program that does what the step asks.

how many is enough?

Open task: any program that does what the step asks.

Stretch and support

Stretch: How many flips are needed, roughly, to be within 0.01 of 0.5 most of the time? Experiment with the board (about 10 000).

Support: Use the board's graph: point to where the line is after 10, 100 and 1000 trials.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Many trials".

The idea: Law of large numbers — observed vs predicted: I can Students observe that relative frequencies become closer to theoretical probability as the number of trials increases, and explain why short-run results may differ from predictions.

What understanding looks like:
- States that relative frequencies approach theoretical probability as trials increase.
- Explains why short-run results can differ significantly from predictions.
- Does not commit the gambler's fallacy (thinking past outcomes affect future probabilities).

Mistakes students often make here:
- A student thinks heads is 'due' (Zara's mistake).
- A student thinks large experiments give the probability exactly.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Running experiments and recording results), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Many trials".

Exit ticket

After 40 spins a spinner gave 18 reds; after 4000 spins, 1230 reds. Which estimate of P(red) is better, and what is it?

The 4000-spin one: 1230 ÷ 4000 ≈ 0.31. More trials give a more reliable estimate.

What each step is for

• 10 or 1000?: Committing to which experiment lands closer to half shows whether they expect larger samples to be more reliable.
• watch it settle: Running 10 flips, then a thousand, on one graph shows the early jumps and the settling: relative frequency approaches the probability as trials grow.
• settling, not balancing: Relative frequency settles because each new trial counts for less as the total grows, not because earlier results are balanced out. Zara's mistake thinks the coin remembers.
• one change at a time: One change at a time: a dice instead of a coin, a mystery bag estimated from data, the size of the error, then the edge case of a long run.
• many trials: Relative frequency, estimating from experiments and judging sample size, mixed; the gambler's slip planted again; the review keeps expected counts fresh.
• how many is enough?: Running trials until the estimate is within 0.01 of the probability makes the law of large numbers concrete: closeness costs many trials.

Where it leads

This completes the Year 7 chance descriptors (AC9M7P01, AC9M7P02).

Where it leads: complementary events, two-step experiments and simulations of compound events (Year 8).