Mathematics from zero
Probability: predict and test a chance game
A rule you can only state is a rule you do not yet trust. Here you will predict the probabilities of a small chance game with the unit’s three moves — favourable over total, multiply, add, complement — and then actually play it dozens of times to see whether reality lands where your arithmetic said it would.
Turn the unit’s rules into a working loop: list a game’s outcomes, compute each probability by hand, run the game many times, and compare the counts you observed against the probabilities you predicted.
Design a simple two-stage chance game (two coins, two dice, or a bag of coloured tokens), predict the probability of every key outcome using only this unit's rules, then run the game at least 40 times and check your predictions against what actually happened.
- A complete outcome list for your game, with a one-line check that the outcomes are equally likely.
- Three predicted probabilities, each labelled with the rule used — multiply, add, or complement — and the arithmetic shown.
- A results table with three rows: event, predicted probability, observed fraction from at least 40 runs.
- One short paragraph: for any event whose observed fraction differs from your prediction, explain why a small number of runs does not match the exact probability, and what you would expect if you ran the game 1000 times instead.
- Add a fourth event that mixes rules — e.g. 'at least one 6 in two rolls' — and show it is easiest to compute as the complement of 'no 6 at all' (1 minus 5/6 × 5/6).
- Make one stage dependent: draw two tokens WITHOUT putting the first back, recompute the second-draw probability, and explain how the numbers changed from the independent version.
- Increase your runs to 100 or more and show that the observed fractions move closer to your predicted probabilities as the number of runs grows.
This is the loop behind every probability claim: list the outcomes, compute by favourable over total, combine with multiply for ‘both’, add for ‘either exclusive’, and the complement for ‘not’ — then test against reality. Seeing 40 runs land near your predictions, and drift even closer at 100, turns the rules from things you memorised into things you trust.