Theoretical probability = number of favourable outcomes ÷ total equally likely outcomes. A fair die gives P(6) = \frac{1}{6}.
Relative frequency (experimental probability) = number of times the event happened ÷ total number of trials.
Probabilities lie between 0 (impossible) and 1 (certain), and all outcomes sum to 1: P(A) + P(\text{not } A) = 1.
Relative frequency and the law of large numbers
Over many independent trials with a stable underlying probability, relative frequency tends to settle near that probability. More trials generally reduce random variation, but do not guarantee a closer estimate at every step or remove systematic bias.
A spinner landing on red 18 times in 50 spins has relative frequency \frac{18}{50} = 0.36.
Expected outcomes
Expected number = probability × number of trials. If P(\text{red}) = 0.2 and there are 150 trials, expect 0.2 \times 150 = 30.
Expected values are averages; actual results vary.
Expected frequencies
Bias and fairness
If the relative frequencies are very different from equal values over many trials, the die or coin may be biased.
Compare experimental results to the theoretical probability to judge fairness.
Estimating probabilities
Use data (e.g. past weather) to estimate probability when outcomes are not equally likely.
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