Experimental probability and simulation

Experimental vs theoretical probability

• Theoretical probability is predicted from a model, assuming you know how the situation behaves (a fair die gives $P(\text{six}) = \tfrac{1}{6}$).
• Experimental probability (or relative frequency) is measured from data — actually doing trials and counting:

$$\text{experimental probability} = \frac{\text{number of times it happened}}{\text{number of trials}}$$

• Use the experimental value when there is no fair model — e.g. a bent coin, a drawing pin, or a real-world event whose true chance is unknown.

The long-run relative frequency

• The two probabilities usually differ for a small number of trials, purely because of chance (natural variation).
• As the number of trials increases, the experimental probability tends to settle down and get closer to the theoretical value. This is why a fair die rolled thousands of times gives a six close to $\tfrac{1}{6}$ of the time.

Simulation

• A simulation copies a real situation using random outcomes (dice, spinners, or random digits) when the real experiment is hard or slow to run.
• To design a good simulation:
  • Map random outcomes to the real event (e.g. "digits 0–6 = bus on time, 7–9 = late", to represent 70% on time).
  • Run many trials and record each result.
  • Estimate the probability as the proportion of successful trials — more trials give a more reliable estimate.