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Probability lab

Coins, dice, spinners and urns: compare theoretical probability with a random simulation and watch the frequency curve approach it.

Formula

P(A) = \frac{m}{n},\quad P(A) \approx \frac{n_A}{n}

Classical probability is m favourable outcomes over n equally likely ones; an experiment estimates it by the frequency n_A / n, where n_A counts how often A happened.

ExperimentCoins or diceTrials1000Random seed

The random numbers come from a fixed seed, so the same input always gives the same run; change the seed to repeat the experiment.

Presets

Most likely outcome1 heads

Theoretical probability0.5

Observed frequency0.515

n1000

Theoretical probability0.5

Observed frequency0.515

Observed mean0.971

Theoretical mean1

Frequency settling towards the probability

Outcome	Count	Frequency	Theory
0 heads	257	0.257	0.25
1 heads	515	0.515	0.5
2 heads	228	0.228	0.25

Equally likely outcomes: 4
Most likely outcome: 1 heads
Favourable outcomes: 515
Frequency minus probability: 0.015

Steps

4 / 4

1Formula
$P = \frac{m}{n}$
Start from the definition of classical probability.
2Substitute
$n = 4$
Count all equally likely outcomes.
3Solve
$P(\text{1}) = 0.5$
Divide the favourable count by the total.
4Conclusion
$f_{1000} = 0.515$
Then run the experiment and compare frequency with theory.

Concept

Probability measures how likely an event is and always lies between 0 and 1. A certain event has probability 1 and an impossible one 0.

Frequency is what happened; probability is the theoretical value. With few trials the frequency jumps around, and with many it settles near the probability.

Going further

Counting needs equal likelihood: Dividing favourable by total only works when every basic outcome is equally likely. A loaded die or an unbalanced coin breaks the method.

Common mistakes

List all equally likely outcomes first, then pick the favourable ones, instead of reaching for a formula.
Two runs giving different frequencies is normal; compare the trend, not a single number.

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