Binomial Experiment

Many experiments have the following form:

The experiment has $n$ smaller experiments called trials, where $n$ is fixed in advance.
Each trial only has one of two outcomes, success $S$ or failure $F$ .
The trials are independent; one trial's results don't influence another's.
The probability $p$ of success is constant between trials.

Binomial Experiment

An experiment for which the above 4 conditions apply - a fixed number of dichotomous, independent, homogeneous trials.

For example, if a fair coin is tossed $n$ times, say $H$ is our success and $T$ is failure. We know $p = .5$ . This coin toss experiment would be a binomial experiment.

5% population rule

Consider sampling without replacement from a dichotomous population of size $N$ . If the sample size (number of trials) $n$ is at most $5 %$ of the population size, the experiment can be analyzed as though it were a binomial experiment.

Why is this the case? As an example suppose $500, 000$ trucks have been sold over the last 5 years and that $350, 000$ of the owners are satisfied with their vehicle. A sample of $10$ owners is chosen (without replacement). Regard each selected owner as constituting a trial, with the $i$ -th trial labeled $S$ if the owner is satisfied. Although sampling without replacement would again appear to invalidate (3: independence of the trials), the important note is that the population being sampled here is very large relative to the sample size. Namely:

P (S on second | S on first) = \frac{349, 999}{499, 999} = .6999 \dots \approx .7000 \dots

and:

P (S on tenth | S on first nine) = \frac{349, 991}{499, 991} = .699995 \approx .7000

so they're so close that we can regard this as a Binomial Experiment with $n = 10$ and $p = .7$ or whatever it's given to be.