Repeated chance experiments and simulations

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.

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.

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.

One run, step by step

A single experiment is the building block of everything here. Take one fixed run of ten coin flips and reveal it one flip at a time. After each flip you recompute the running relative frequency of heads, the heads counted so far divided by the flips so far. Early on this value swings wildly, from 1.000 after a single head down past 0.500, because one result is a large share of a tiny sample. As the flips accumulate, each new outcome moves the running value less, and it begins to settle near one half.

Simulating many trials at once

A simulation lets you carry out a large number of trials in moments. Imagine rolling a fair die sixty times. Probability predicts each of the six faces about ten times, since one sixth of sixty is ten. A simulated run will not give exactly ten of each, but the observed counts cluster around ten, and the frequency bars sit close to the predicted line. Comparing the bars with that prediction is the core habit of experimental probability.

Why bigger simulations matter

The real payoff of large simulations is precision. Run a small batch of coin flips and the observed proportion of heads can sit well away from one half. Run a larger batch and that proportion lands closer to 0.5. Stepping a batch from ten flips up to five hundred, the gap from one half shrinks from about 0.2 to a few thousandths. This is why digital tools matter: they make the long run visible, and the long run is where chance reveals its underlying pattern.

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.