Probability is one of the most counter-intuitive topics in the maths curriculum, and it's also one of the most teachable — because every concept has a physical demonstration. A coin, a die, and a way to generate random numbers are enough to take students from "what are the chances?" to a genuine understanding of theoretical versus experimental probability. This guide lays out a lesson arc you can run with a class of any size, using free online tools when you don't have enough physical coins and dice to go around.
Start with prediction, not formulas
Before introducing any vocabulary, ask the class to predict. "If I flip this coin ten times, how many heads?" Write the guesses on the board. Then flip a coin ten times and record the result. The gap between what students expect (exactly five heads) and what actually happens (often three, four, six, or seven) is the hook for the entire unit. It creates a question students want answered: why doesn't the experiment match the prediction?
Activity 1: Coin flips and the law of large numbers
Have each student (or pair) flip a coin 10 times and tally heads and tails.
Collect every group's results on the board and add them into a class total.
Compare a single group's ratio (often far from 50/50) with the whole-class ratio.
Discuss why the combined result lands much closer to 50% — the law of large numbers in action.
If you don't have enough coins, an online coin flipper lets every student run a batch of flips instantly and even flip 100 times at once, which makes the convergence toward 50% dramatic and obvious. The key lesson: a fair coin doesn't promise an even split in a few flips, only over many.
Activity 2: Dice and combined outcomes
Roll two dice and ask students to predict which total comes up most often. Many guess that all totals from 2 to 12 are equally likely. Rolling repeatedly — or using a dice roller to generate hundreds of rolls quickly — reveals the famous triangular distribution, where 7 appears far more often than 2 or 12. This is the perfect moment to introduce sample space: there's only one way to roll a 2 (1+1) but six ways to roll a 7.
Have students list every combination that makes each total. Seeing that 7 has six combinations while 12 has one turns an abstract result into something they discovered themselves.
Activity 3: Address the gambler's fallacy
After a run of, say, five heads in a row, ask: "Is tails more likely now?" Many students will say yes. This is the gambler's fallacy, and it's worth confronting directly. Each flip is independent — the coin has no memory — so the next flip is still 50/50 regardless of the streak. A coin-streak game, where students predict the next flip and try to build a run, makes this vivid: long streaks happen, and they never make the next flip any easier to call.
Bringing it together: theoretical vs experimental
Close the unit by formalising what students have seen. Theoretical probability is what the maths predicts (1/2 for heads, 1/6 for a particular die face). Experimental probability is what actually happened in their trials. The two move closer as the number of trials grows. Because students arrived at this through their own experiments, the definitions land as a summary of something they already understand rather than a rule to memorise.