A Bernoulli Experiment is a random experiment which results in one of the two possible outcomes that are mutually exclusive and mutually exhaustive .
The classical example of Bernoulli trial is a coin toss . A single coin toss has two possible outcomes namely the head and the tail. These two outcomes are mutually exclusive since only one of them can result in a trial. They are mutually exhaustive because there is no third outcome.
While describing a Bernoulli trial, any one of the two possible outcomes can be conveniently designated as "success" and the other one as "failure". If \( p \) is the probability of "success", then \( 1-p \) will be the probability of "failure". Since these two outcomes are mutually exhaustive, their probabilites should add up to 1, which is the case here. For example, in the case of a coin toss experiment, if "head" is designated as success, the probability of success \( \small{p = \dfrac{1}{2} }\), and the probability of "failure"(tail) is \(\small{ 1-p = 1-\dfrac{1}{2} = \dfrac{1}{2} }\).
When a Bernoulli experiment is repeated many times independently such that the probability of success (and hence the probability of failure) remains the same for all the experiments, they constitute a sequence of Bernoulli trials. Many real life problems in statistics can be viewed as Bernoulli trials, as shown below:
Suppose we conduct \( n\) bernoulli trials that result in the sequence of successes and failures. Out of n trials, let x of them result in a success and the remaining (n-x) result in a failure. What is the probability that this happens? Let \(p\) be the probability of success and \(1-p\) be the probability of failure for a single trial. Since the n trials are independent of each other, their combined probability is the product of their individual probabilities. There are x independent trials each with a probability p and (n-x) trials each with probability (1-p). If the order of successes and failures do not matter, they can separtely computed. One can write, \( \small{P(n~trials) = [p \times p \times p \times .....(x~times) ] \times [(1-p) \times (1-p) \times .....(n-x)~~times) ] = p^x(1-p)^{n-x} }\)
We arrive at the important expression for the Bernoulli trials:
If p is the probability of success in a Bernoulli trial, then the probability for a sequence of trials resulting in a total of \(x \) successes and \((n-x)\) failures occuring in any order is given by ,