Bernoulli's Scheme

Bernoulli’s Scheme


(named for Jakob Bernoulli), one of the basic mathematical models used in probability theory for the description of independent repetitions of experiments. Bernoulli’s scheme assumes that there exist a certain experiment S and a random event A associated with it (a typical example: S is the tossing of a coin, and A is the occurrence of heads). Let n independent repetitions of 5 be carried out. For each performance of 5, the event A occurs (a success, so to speak) with probability ρ (in the proposed example, ρ = 1/2) and does not occur (failure) with probability q = 1 - p. Thus, Bernoulli’s scheme is defined by two parameters—n and p. The probability of one or another number of successes is given by the binomial distribution. Major principles of the theory of probability (for example, the law of large numbers) were discovered based on the example of Bernoulli’s scheme. The replacement of the condition of independence of the experiments in Bernoulli’s scheme by the condition of the dependence of each experiment on only the one immediately preceding leads to another major model of probability theory—the Markov chain.