Repeatable experiments

In practice, experiments rarely have outcomes that we can argue must be equally likely. In any such experiment, a different definition must be used for the probability of an event.

We can often imagine repeating the experiment many times. If the experiment is repeated indefinitely in essentially the same way, the probability of an event can be defined as the long-term proportion of times that it occurs.

Definition

If an experiment could be repeated in essentially the same way again and again, then the probability of any event, \(E\), is defined to be the limiting proportion of times that the event occurs as the number of repetitions increases.

\[P(E) = \lim_{\text {repetitions} \to \infty} (\text {proportion of times } E \text { occurs})\]

The next diagram illustrates.

Consistency with earlier definition

A theorem called the Law of Large Numbers proves that this definition is consistent with the definition that we gave earlier for experiments with equally likely outcomes. If the probability of an event is \(p\) (using the classical definition) and the experiment is repeated indefinitely, the limiting proportion of times that the event occurs is always \(p\).