This is an excerise for Udacity's AI Programming with Python lesson 7.

A Binomial distribution is a probability distribution that summarizes the likelihood of a binary outcome, such as success or failure, in a fixed number of independent trials. It's characterized by two parameters:

1. Number of Trials (n): The total number of independent experiments or trials.
2. Probability of Success (p): The probability of success in each individual trial.

In a Binomial distribution:

• Each trial is independent, meaning the outcome of one trial does not affect the outcome of another.
• Each trial has only two possible outcomes, often labeled as success (usually denoted as 1) or failure (usually denoted as 0).
• The probability of success (denoted as p) remains constant from trial to trial.
• The number of successes in n trials, denoted as k, can range from 0 to n.

The probability mass function (PMF) of the Binomial distribution gives the probability of observing exactly k successes in n trials, given the probability of success p.

The formula for the PMF of a Binomial distribution is:

[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} ]

where:

• ( \binom{n}{k} ) is the binomial coefficient, representing the number of ways to choose k successes out of n trials.
• p is the probability of success on each individual trial.
• k is the number of successes.
• n is the total number of trials.

Binomial distributions are widely used in various fields, including statistics, finance, biology, and quality control, to model phenomena such as success/failure experiments, coin flips, and the number of defective items in a sample.