A binomial distribution is a probability distribution that describes the likelihood of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant throughout the trials.
The key characteristics of a binomial distribution include a fixed number of trials, two possible outcomes for each trial, a constant probability of success, independent trials, and discrete data.
The binomial distribution is related to the binomial theorem in that it provides a way to calculate the probability of obtaining a specific number of successes in a certain number of trials, while the binomial theorem is a formula used to expand binomials raised to a power.
Some real-life examples of binomial distributions include coin tosses, where the outcome is either heads or tails, and the success rate of a medical treatment, where the outcome is either a cure or no cure.
A binomial distribution is different from a normal distribution in that it describes discrete data with two possible outcomes, while a normal distribution describes continuous data. Additionally, the shape of a binomial distribution is skewed, while the shape of a normal distribution is bell-shaped.