AC9M7P02 · Year 7 · Probability

Repeated chance experiments and simulations

ACARA v9 CONTENT DESCRIPTION “conduct repeated chance experiments and run simulations with a large number of trials using digital tools; compare predictions about outcomes with observed results, explaining the differences”

Knowing the theoretical probability of an outcome is one thing; seeing what actually happens when you try it many times is another. This unit is about experimental probability, found by actually conducting chance experiments and running simulations, and comparing the results with what theory predicted. This year you learn to carry out repeated trials, use digital tools to simulate large numbers of them, and explain why observed results and predictions do not always match exactly.

This is the natural partner to working out theoretical probabilities. Theory tells you what should happen in the long run; experiments and simulations let you watch that long run unfold, and the relationship between the two is one of the most important ideas in all of probability.

Experiments and the long run

When you conduct a chance experiment, such as flipping a coin, you record the relative frequency of each outcome: the number of times it happened divided by the number of trials. With only a few flips, this proportion can be far from the theoretical value; you might get 7 heads in 10 flips, a relative frequency of 0.7 rather than 0.5. But as you keep flipping, something striking happens.

More trials, closer to theory
As the number of trials grows, the experimental frequency settles near the theoretical probability.
With only a few trials, the proportion of heads can swing far from one half. As the number of trials grows into the hundreds and thousands, the experimental relative frequency settles down close to the theoretical probability of 0.5.

The more trials you run, the closer the relative frequency tends to settle toward the theoretical probability. Over thousands of flips, the proportion of heads hugs 0.5 closely. This is why a large number of trials matters, and why digital tools are so valuable: a simulation can run thousands or millions of trials in seconds, letting you see the long-run behaviour that would take hours to produce by hand.

Comparing prediction with observation

With a known probability you can predict results. If a coin is fair, you predict about 50 heads in 100 flips. When you actually run the experiment, you might observe 46 heads and 54 tails. The prediction and the observation are close but not identical, and that small difference is completely normal. It comes from natural variation, the ordinary randomness of chance, not from any error.

Predicted versus observed
Compare what probability predicts with what an experiment actually produces.
Over 100 coin flips, probability predicts about 50 heads and 50 tails. A real experiment might give 46 and 54. The small gap between predicted and observed is normal, and it tends to shrink, in proportion, as the number of trials increases.

Explaining these differences is the heart of the unit. A gap between predicted and observed results does not mean the theory is wrong or the experiment was botched; it reflects the natural variability of random events, which is most noticeable when the number of trials is small. As the number of trials grows, the observed relative frequency moves, in proportion, ever closer to the prediction. Understanding this link between theoretical probability and experimental results, between the short-run surprises and the long-run pattern, completes the picture of how chance behaves and lets you reason sensibly about everything from games and surveys to the simulations that model the real world.

Teaching tip: do a real experiment, then a simulated one. Flip a coin 20 times and note how far the result can stray from 10 heads, then use a spreadsheet or online tool to simulate 1000 or 10000 flips and watch the proportion settle near one half. The contrast between the small and large samples makes the long-run idea unforgettable.

Reassure the student that observed results rarely match predictions exactly, and that this is expected, not a mistake. Emphasise that the gap tends to shrink in proportion as trials increase. This guards against the common belief that a fair coin must give exactly half heads in any short run.

Builds on: Sample space and probabilities of single-stage events (AC9M7P01). That unit assigned theoretical probabilities; this unit tests them with repeated experiments and simulations.
Quick self-check
1. What is a relative frequency in a chance experiment?
2. As the number of trials in an experiment increases, the relative frequency tends to
3. Why are digital tools useful for chance experiments?
4. A die is rolled 60 times. About how many sixes does probability predict?
5. In an experiment, the observed results differ slightly from the prediction. This usually means