What is Binomial Distribution?

Definition

The binomial distribution is a discrete probability distribution used to describe the probability of a certain number of successes in a fixed number of independent trials. Each trial has only two possible outcomes, typically referred to as 'success' and 'failure'. The binomial distribution is defined by two parameters: the number of trials n and the probability of success p in each trial.

Origin

The concept of the binomial distribution dates back to the 17th century, evolving with the development of probability theory. Jacob Bernoulli systematically described this distribution in his work 'Ars Conjectandi', laying the foundation for modern probability theory.

Categories and Features

The main features of the binomial distribution are its discreteness and binary nature. It is applicable in situations where each trial has only two possible outcomes, such as coin tosses or product quality checks. Its probability mass function is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is a binomial coefficient.

Case Studies

Case 1: Suppose a company has a 95% pass rate for its light bulbs. The probability of at least 90 out of 100 bulbs being acceptable can be calculated using the binomial distribution. Case 2: In a market survey, if a product has an 80% satisfaction rate, the probability that 8 out of 10 surveyed people are satisfied can also be calculated using the binomial distribution.

Common Issues

Common issues include misapplying the binomial distribution to non-independent trials or when the number of trials is not fixed. Additionally, misunderstanding the definition of parameter p can lead to incorrect probability calculations.