Experimental Probability

Sometimes the outcomes are not equally likely, so you can't just count cases. A bent coin, a drawing pin or a weighted spinner has no obvious symmetry to lean on. When that happens, you estimate the probability by running the experiment many times and watching how often the outcome shows up — its relative frequency.

\text{relative frequency} = \frac{\text{number of times it happened}}{\text{total number of trials}}

The more trials you run, the closer this fraction settles towards the true probability. A few throws tell you little; thousands tell you a lot.

the relative frequency of an outcome is \dfrac{\text{successes}}{\text{trials}};
it estimates the true probability, and improves as the number of trials grows;
the expected frequency — how many times you predict it will happen — is \text{probability} \times \text{number of trials}.

Reading an experiment

Here are the results of spinning a coloured spinner 50 times. Each bar is a count; divide by the total to estimate the probability of that colour.

For example, red came up 22 times, so the relative frequency of red is \tfrac{22}{50} = 0.44 — our best estimate of P(\text{red}).