Use coins and colored suit categories. Focus on more likely, less likely, equally likely, and simple fractions like 1/2 and 1/4.
Mathematics · Lesson 04
Probability Simulator
Probability often feels surprising because small samples are noisy. This lesson turns chance into something students can see. By running many virtual coin flips, dice rolls, and card draws, they compare what should happen in theory with what actually happens in practice, and watch experimental probability settle toward expected probability over time.
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This lesson is built to work as a fast middle school introduction, a high school statistics warm-up, or a hands-on intervention for students who confuse a short run of outcomes with true probability. The simulator does the repetition for them so class time can focus on noticing patterns, making claims, and defending them with data.
Students Should Leave Able To
- distinguish theoretical probability from experimental probability.
- explain why small samples can look unfair even when a process is fair.
- use repeated trials to argue for the law of large numbers.
- read percentages, fractions, and counts from a simulation summary.
5-Minute Start
- 1Ask students whether 10 coin flips should always split into 5 heads and 5 tails.
- 2Run 10 flips, then 100, then 1000, and have students compare the percentages.
- 3Introduce the phrase "closer in the long run, not guaranteed in the short run."
Compare theoretical and experimental probability, calculate percentages, and explain why samples of 10 and 1000 tell different stories.
Discuss convergence, sample size, bias, independence, and why simulation is useful when an experiment is slow, expensive, or impractical.
Interactive Lab
Run the Probability Simulator
Choose an experiment, set the number of trials, and compare the observed results to the expected probabilities. Use the checkpoint table to show students how the data stabilizes as the sample gets larger.
Build a discrete probability space using outcome labels and whole-number weights. The simulator normalizes the weights for you, then uses the same engine as the preset experiments.
Start with two to six outcomes. Integer weights like `3` and `7` work better than typing decimal probabilities.
Current model
A fair coin has two equally likely outcomes, so heads is expected about half the time in the long run.
Teacher move
Before clicking run, have students predict whether the experiment will look "fair." Afterward, ask whether their claim is based on counts, percentages, or both.
The engine decides the result first. The animation only reveals that already-chosen outcome.
Tracked event
Heads
Expected
50.0%
Observed
--
Difference
--
Outcome Distribution
Convergence Graph
Expected probability
Observed rate
Short runs wobble more. Larger samples usually settle closer to the expected line.
Checkpoint Comparison
| Trials | Observed event rate | Gap from expected |
|---|
Run a simulation to see how experimental probability changes as the sample grows.
Teach the Math
Theoretical probability is what mathematics predicts before we experiment. For a fair coin, the probability of heads is 1/2. For a fair six-sided die, the probability of rolling a 6 is 1/6. For a draw where each suit is equally likely, the probability of hearts is 1/4.
Experimental probability is what actually happened in your trials. If heads appears 47 times out of 100 coin flips, the experimental probability is 47/100, or 47%.
Core idea
The law of large numbers says that as the number of trials gets larger, the experimental results tend to move closer to the theoretical probability. It does not say the results become perfect, and it does not say short runs have to look balanced.
One single attempt in an experiment, like one flip or one roll.
The fraction or percent of times an outcome appears in the data collected so far.
One result does not change the probability of the next result in the same model.
A streak does not force the next outcome to compensate. That is the gambler's fallacy.
Suggested Lesson Sequence
This structure fits a 40 to 50 minute period, but it can also expand into a two-day lesson by giving groups different experiments and requiring a short written claim backed by simulation data.
Ask: "If I flip a coin 10 times, should I expect exactly 5 heads?" Let students commit to a prediction before seeing any data.
Run the simulator at 10, 100, and 1000 trials. Compare the observed rate at each checkpoint and name the pattern out loud.
Pairs choose one experiment and complete a claim-evidence-reasoning response using counts, percentages, and the checkpoint table.
Compare groups. Which experiment looked most stable early? Which needed more trials? Why do percentages matter more than raw counts?
Students explain why a fair process can produce uneven short-run data and how larger samples help.
Discussion Prompts and Teacher Notes
Use these questions to keep the lesson focused on reasoning, not just button clicking. Students should make a claim before the run, revise after the data, and connect what they saw to a probability principle.
What looks more convincing: 6 heads in 10 flips, or 510 heads in 1000 flips? Why?
If two classes run the same simulation, should their counts match exactly? Should their percentages be close?
Why does a difference of 5 feel huge in 10 trials but tiny in 1000 trials?
How is simulation useful when a real experiment would take too long or cost too much?
Instructional note
If students say the simulator is "wrong" because the outcomes are not perfectly balanced, redirect them to percentages and sample size. The tension between fairness and variation is the whole lesson.
Assessment and Extension
Show 8 heads in 10 flips and ask whether the coin must be unfair. Students should answer no and justify using sample size.
"How did the simulation support the law of large numbers? Use one exact percentage from your run as evidence."
Have students design a new experiment such as a spinner, raffle, or weather model and predict its long-run behavior before coding it.
Connect to genetics, radioactive decay, or particle detection where outcomes are probabilistic but patterns emerge over many observations.
Introduce simulation as a tool when direct calculation is hard, especially in AI, games, and Monte Carlo methods.
Discuss why polls and surveys need enough responses, and why small samples can create misleading headlines.