Bernoulli Trials

Data Science, AI and ML
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Consider an experiment and an event A within the sample space. We say the experiment is a
success if an outcome from A occurs and failure otherwise. Let us consider the following examples:
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In typical applications we would repeat an experiment several times independently and would be
interested in the total number of successes achieved, a process that may be viewed as sampling
from a large population. For instance, a manager in a factory making nuts and bolts, may devise
an experiment to choose uniformly from a collection of manufactured bolts and call the experiment
a success if the bolt is not defective. Then she would want to repeat such a selection every time
and quantify the number of successes.

Bernoulli Trials
We will now proceed to construct a mathematical framework for independent trials of an experiment where each trial is either a success or a failure. Let p be the probability of success at each trial. The sequence so obtained is called a sequence of Bernoulli trials with parameter p. The trials are named after James Bernoulli (1654-1705). We will occasionally want to consider a single Bernoulli trial, so we will use the notation Bernoulli(p) to indicate such a distribution. Since we are only interested in the result of the trial, we may view this as a probability on the sample space S = {success, failure} where P({success}) = p, but more often we will be interested in multiple independent trials.