Over a few trials the relative frequency jumps about; over many it settles close to the probability. That is why more trials give a better estimate, and why a run of tails does not make heads 'due'.
Flip a coin ten times and you might get 7 heads. Flip it ten thousand times and you will be very close to half. Why does more help?
In Predict and test you saw that a short experiment rarely matches the expected count. This lesson asks what happens as the number of trials grows.
The relative frequency (count ÷ trials) jumps about over the first few trials, then settles closer and closer to the probability. This is called the law of large numbers. It is why a large experiment (or a simulation) gives a good estimate of a probability you cannot work out from a 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.
10 or 1000?
Picture 10 flips. How likely is exactly 5 heads?
TryTry it on the board and compare the result with your prediction.
Watch it settle
Where is the line after 10 flips? After 1000? Press Start again and run it again: does the start look the same?
Try it yourselfYour edits are saved in this browser.
Run 10 flips, then keep going to at least 1000. Watch the line. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Settling, not balancing
Here is why more trials help.
Worked exampleBefore each step, say what you think comes next.
107=0.7Ten flips: one result changes the fraction by 0.1.
107→118≈0.73One more head moves it a lot.
1000501→1001502≈0.5015After 1000 flips, one more head barely moves it.
each flip: P(H)=21The coin has no memory: nothing is "due".
Now find the relative frequency after 1000 flips.
Read Zara's reasoning. Does the coin know what happened before?
After 1000 flips, a coin has shown 482 heads. What is the relative frequency of heads? (A decimal.)
Zara thinks about her next flip. Click the line where her reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Run 1000 at a time. P(6) is 1/6 ≈ 0.167.
Need a hint?
You cannot see the bag. Run at least 1000 draws (each counter goes back in).
Another hint
Relative frequency of red × 10 estimates how many of the 10 counters are red.
Try it yourselfYour edits are saved in this browser.
Roll at least 2000 times. Does the relative frequency of sixes settle near 61≈0.167?
Use the board. This ticks when you get there.
A mystery bag holds 10 counters, red and yellow. Draw (with replacement) at least 1000 times and estimate how many are red.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Many trials
A drawing pin landed point-up 312 times in 800 throws. Estimate the probability it lands point-up. (A decimal.)
Using your estimate of 0.39, how many point-up landings would you expect in 200 throws?
Review: how many heads do you expect in 300 flips of a fair coin?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
How many is enough?
How many flips did it take? Compare with others.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Run 1000 at a time.
Another hint
Keep going until the relative frequency of heads is between 0.49 and 0.51.
Keep flipping until you have at least 1000 flips and the relative frequency of heads is within 0.01 of 0.5.
Use the board. This ticks when you get there.
For teachers
Students run hundreds and thousands of trials and watch the relative frequency settle towards the probability (the law of large numbers), use large experiments to estimate unknown probabilities, and reject the idea that outcomes are 'due' (AC9M7P02, node 2). About 40 minutes.
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'.
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
Predict: take a show of hands; many students choose 'equally likely'.
Run: ask students to press Start again and compare the first ten flips of different runs.
Investigate: show the 7/10 → 8/11 and 501/1000 → 502/1001 calculation on the board.
Modify: collect the class's estimates for the mystery bag; reveal 3 red, 7 yellow.
Make: collect how many flips each student needed.
When students get stuck
A student thinks heads is 'due' (Zara's mistake). Ask: How would the coin know what happened before? Fix: Each flip is still 1/2; long runs happen, and get swamped by many more trials.
A student thinks large experiments give the probability exactly. Ask: After 1000 flips, is it exactly 0.5? Fix: Close, not exact: closer with more trials.
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.
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).
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.
Now you can
Describe how the relative frequency changes as the number of trials grows.
Use a large experiment to estimate a probability.
Explain why a run of one outcome does not make another one 'due'.