Coin laboratory
Compare predictions with runs of 10, 100, and 1,000 flips.
A short evidence lab
Think first. Test second.
Probability is a map of what could happen. Your experiments are evidence, not promises.
Dice investigation
See why two-dice sums do not all have equal chances.
Fair-game designer
Use expected points to test a game’s balance.
About 10–15 minutes. You can skip or revise every prediction. Progress stays in this page's memory only; refreshing or Reset all clears it.
What will you investigate?
A trial is one toss or roll. Theoretical probability is the chance given by a model. Observed frequency counts what actually happened in your experiment; divide by the number of trials for an observed proportion.
Example: 7 heads in 10 tosses is an observed 70%. A fair coin's theoretical chance is still 50%. Try making your own comparison.
Investigation 01
The coin laboratory
Observed frequency can wiggle, especially in a small sample. A fair coin still has a theoretical 50% chance on every flip.
Heads—
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Tails—
—
Detective note
Run the experiment to compare observed proportions with the model’s theoretical value.
Each toss is independent: earlier heads do not make tails due. Even after ten heads, the next toss of a fair coin has a 1/2 chance of heads. Larger samples often have proportions closer to theory, but no single result or steady improvement is guaranteed.
Hint: compare runs fairly
Keep the same coin model, change the sample size, and repeat. Compare percentages, not raw counts. A result can move further from theory before moving closer.| Trials | Heads | Observed | Theory |
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Investigation 02
The dice investigation
One die has six equally likely faces. With two dice, middle sums have more combinations than extreme sums.
| Outcome | Count | Observed | Theory |
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Why are sums different?
Two independent fair dice have 36 equally likely ordered pairs. Sum 2 has only (1,1): chance 1/36. Sum 7 has (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): chance 6/36. The eleven sums are not equally likely.
Hint: count pairs, not totals
A first die of 1 and a second die of 6 is a different outcome from 6 then 1. Both produce the same sum.Investigation 03
Design a fair points game
Fair here means equal expected points per round. Points are a model, never money.
Write the rules
Each round has exactly one scoring side. A scores its reward with the selected chance; otherwise B scores its reward. The other side gets zero. Rounds are independent.
Expected points per round
Side A—
Side B—
Expected points = chance of scoring × reward.
Change the rules to test the model.
A points —B points —
Hint: a smaller chance can balance a bigger reward
If A scores 6 points with a 25% chance and B scores 2 with a 75% chance, each expects 1.5 points per round.Fresh challenge
Can you spot the fair game?
Evidence so far
Your detective report
Next practice: design a game with unequal scoring chances but equal expected points. Explain it using arithmetic, then test 100 rounds and explain any score difference.