Probability and population proportion

When sampling from a finite population, the probability of any 'event' is the proportion of population values for which that 'event' happens. For example, the probability that a randomly selected household from a town contains more than two adults equals the proportion of households of that size in the town.

The same definition can be used for infinite populations (distributions), even if the population is abstract. When selecting one value from the population,

The probability of any value or range of values equals the proportion of these values in the population.

Probability and long-term proportion

There is an alternative but equivalent way to think about probability when it is possible to imagine repeatedly selecting more and more values from the population (e.g. repeating an experiment again and again).

The probability of any value or range of values is the limiting proportion of these values as the sample size increases.

The fact that the sample proportion always stabilises at the probability (i.e. the population proportion) is called the law of large numbers.

Sex of chickens

The sex of a newly hatched chicken is a categorical value (male or female). The randomness of the chicken's sex can be modelled as being a value that is randomly sampled from an abstract infinite population in which a proportion of values are male and the rest are female.

The probability that one baby will be male is the proportion of male values in this underlying population.

Alternatively, we can imagine a sequence of more and more chickens being hatched. The probability of one chicken being male is also the limiting proportion of males in this (imaginary) sequence of hatchings.

These are two different ways to think about the probability, but the value is the same.

Law of large numbers

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.