Математика. Материалы курса

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Repeated Independent Experiments and the Bernoulli Formula

Let us consider the Bernoulli scheme. Let us have the space of elementary outcomes of some experience Ω and event A for this experience. We are going to repeat this experience many times. We are going to consider the situation successful, if the event A has occurred, and it is a failure if the event A has not occurred. The letter p denotes the probability of the event A and the letter q denotes the probability of the event no-A (the probability of a failure). Then we will call the Bernoulli scheme a sequence of independent identical experiments, in each of which we are going to consider success and failure.

Let’s consider the outcomesup to the first success. Most often, they consider outcomesup to the first success, or a fixed number of outcomes.

Case 1. The Bernoulli scheme before the first success.

LetА_ibe a successful outcome occurring in the i-thexperiment and the letter τ is the number of the first successful outcome. Then τ=k indicates that exactly k tests were carried out before the first success. This probability can be represented as the probability of the product, we receive a failurein the first k-1 experiments, and only in the k-thexperiment there is a success. Since the experiments are independent, the probability of the product of these experiments will be equal to the product of probabilities. Note, the probability of everyА_i is always equal to p, and the probability of negation is always equal to q. Therefore, the probability of the event τ=k is equal to q^k-1. You can use this formula. The important property of such tests is that if we have already carried out several unsuccessful experiments, we can forget about them. That is, the conditional probability that more than n+k experiments will be carried out, provided that n experiments have already failed, will be equal to the probability that we will have more k unsuccessful experiments, that is, without any background. If we consider the repetition of n experiments, then there is no way to do this.

Let's say we had n independent experiments, and the probability of success is still p, and the probability of failure is q. Let the letter ξbe the number of successful experiments out of n, and the letter k be the number itself. "ξ=k" means that the event has occurred; there are exactly k successful attempts.Then we can calculate the probability of this event using the formula given in the slide, where n by k is the number of combinations of n by k.

Let's prove this formula. For convenience, we will write this probability as P_p(n, k).

Let А_i be "there was success in the i-th experiment", then the events "ξ=k" can be represented as the sum of incompatible events, where among the sequence of m outcomes there are exactly k successes and n - k failures. That is, we get only n by k terms. In other words, any k experiments can be successful, and the remaining n-k can be unsuccessful. Then the probability of the event "ξ=k" can be represented as the sum of the probabilities of these products. Let’s find the probability of the very first product.

Since the experiments are independent, the probability of this product is equal to the product of probabilities.

In this case, for all А_i, their probability is p, and for all non-А_i probabilities are q.Therefore, the probability of the first product is p^k *у^n-к_.

Note that, sincethere are always exactly k successes and n - k failures, then the probabilities of terms are identical.Therefore, the probability of their sum is equal to the sum of their probabilities, the same probabilities; and itis calculated by the Bernoulli formula given earlier.

Let us consider the properties of the Bernoulli formula. The first one. Let'sconsider what the sum of all probabilities is when k passes through all the values from 0 to 1. That is, we have one success, or zero success, two or three successful outcomes, and so on, or k successful outcomes. It is clear that at least some number of successful outcomes will occur. So these events "ξ=0", "ξ=1", " ξ=2 " and so on, "ξ=n" are a complete group of incompatible events, which means that their sum is a reliable event.Since they are incompatible, the probability of the sum is equal to the sum of their probabilities. Thus, the sum of the probabilities of these events is equal to the probability of a reliable event, i.e. 1.

Another property that comes in handy when solving problems is the most probable number of successes. For example, we have some experiment.

We need to find the number of times the event is most likely to occur among a series of n experiments.

We use the formula for neighboring probabilities, that is, we represent p with parameters n and k + 1 through p with parameters n and k, similarly we represent parameter p with parameters n and k through p with parameters n and k-1, and therefore, we represent p with parameters n and k-1. Since k₀ is the most probable, this probability is greater for it than the probabilities of its neighbors, so we can write the system. Having solved this system, we get the following system of inequations.

Note that the interval from np-p to np + p has a length of 1, which means that there can be either one integer or two integers in this interval.