binomially

Binomially refers to something that can be described using a binomial or that follows a binomial distribution. A binomial is a mathematical expression that consists of two terms, typically connected by a plus or minus sign. For example, x + y is a binomial, as is 3a - 2b. In algebra, binomials are often used in operations like multiplication and expansion, such as when squaring a binomial like (x + y)^2, which expands to x^2 + 2xy + y^2.

In probability and statistics, the term "binomially" is most commonly associated with the binomial distribution. A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success is constant for each trial. For instance, if you flip a fair coin 10 times, the number of heads you get would follow a binomial distribution. The probability of getting exactly k successes in n trials, with probability of success p, is given by the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient. Therefore, when data is said to be distributed binomially, it means it adheres to the characteristics of this specific probability model.