BernoulliVersuchen

BernoulliVersuchen, also known as Bernoulli trials, are a basic concept in probability theory. They consist of a sequence of independent experiments, each yielding one of two outcomes: success or failure. In every trial, the probability of success is a fixed p with 0 < p < 1, and the probability of failure is q = 1 − p.

Let Xn denote the number of successes in n trials. Xn follows a binomial distribution with parameters n and p. The probability of observing exactly k successes is P(Xn = k) = binomial(n, k) p^k (1 − p)^{n−k} for k = 0, 1, ..., n. The distribution captures the likelihood of different counts of successes across the n trials.

The mean and variance of Xn are E[Xn] = np and Var[Xn] = np(1 − p). A single Bernoulli trial is a Bernoulli random variable with parameter p. The sum of independent Bernoulli variables yields the binomial distribution, and a sequence of Bernoulli trials with the same p is called a Bernoulli process.

Related concepts include the geometric distribution, modeling the number of trials until the first success, and the negative binomial distribution, counting the number of trials needed to achieve a fixed number of successes. Bernoulli trials are widely used to model simple yes/no processes such as coin flips, pass/fail quality checks, or binary outcomes in A/B testing.

History and naming trace to Jacob Bernoulli, who analyzed such experiments in Ars Conjectandi (1713). The framework remains a foundational element in probability theory and statistics.