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. There are no equally likely outcomes in the following experiments.

In each case, we could, in principle, repeat 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.

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

Consistency with earlier definition

There is a theorem called the Law of Large Numbers that 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 \(p\).

Illustration: Sex of chickens

The diagram below illustrates the fact that a sample proportion tends to a limit as the sample size increases. (The limit is the probability.) Imagine recording the sex of a sequence of newly hatched chickens in a poultry farm.

Click Find new value a few times to observe the sex of a sequence of chickens. When only one chicken has been observed, the proportion of females must be either 0 or 1, but after 20 have been observed, the proportion should be somewhere near 1/2.

Continue observing additional chickens until about 1000 have been recorded. By this time, the proportion of females will have stabilised.

(Hold down the button Find 10 values to speed up the simulation.)

If we carried on infinitely long, the proportion would stabilise at a value that we call the probability of a chicken being female.