4.1. Private theorem on repetition of experiments

In the practical application of probability theory, one often encounters problems in which the same experience or similar experiments are repeated several times. As a result of each experience, some event may or may not appear. , and we are not interested in the result of each individual experience, but in the total number of occurrences as a result of a series of experiments. For example, if a group of shots is made for the same goal, we are usually not interested in the result of each shot, but in the total number of hits. In such tasks, it is required to be able to determine the probability of any given number of occurrences of an event as a result of a series of experiments. Such tasks will be considered in this chapter. They are solved very simply in the case when the experiments are independent.

Several experiments are called independent if the probability of a particular outcome of each of the experiments does not depend on what outcomes other experiments had. For example, several consecutive coin toss are independent experiments. Several consecutive draws of a card from the deck are independent experiments, provided that the removed card is returned to the deck each time and the cards are shuffled; otherwise it is dependent experiences. Several shots are independent experiments only if the aiming is performed anew before each shot; in the case when the aiming is performed once before the whole shooting or is continuously carried out in the process of shooting (shooting by a queue, bombing by a series), the shots are dependent experiments. Independent experiments can be performed under the same or different conditions. In the first case, the probability of an event from experience to experience changes. The private theorem relates to the first case, and the general theorem on the repetition of experiments to the second. We will start with a particular theorem, as more elementary. First of all, consider a specific example.

Example. Three independent shots are taken at the target, the probability of hitting it with each shot is equal to . Find the probability that with these three shots we get exactly two hits.

Decision. Denote the event is that exactly two shells hit the target. This event can occur in three ways:

1) a hit on the first shot, a hit on the second, a miss on the third;

2) hit on the first shot, miss on the second, hit on the third;

3) a miss at the first shot, a hit on the second, a hit on the third.

Therefore, the event can be represented as the sum of the works of events:

,

Where - hits at the first, second, third shots respectively, - a miss at the first, second, third shots.

Considering that the three listed event options incompatible, and the events included in the works are independent, according to the theorems of addition and multiplication we get:

,

or denoting ,

.

Similarly, by listing all possible options in which an event of interest to us may appear a given number of times, the following general task can be solved.

Produced by independent experiments, in each of which some event may or may not appear ; probability of occurrence in each experience is equal and the probability of non-occurrence . Required to find the probability that event in these experiences will appear exactly time.

Consider an event that event will appear in experiences exactly time. This event can be implemented in various ways. Decompose event for the sum of the products of events consisting in the occurrence or non-occurrence in a separate experience. We will denote occurrence of an event in the i-th experience; - non-event in the i-th experience.

Obviously, each version of the event (each member of the sum) must consist of m occurrences and non-appearances, i.e. of events and events with different indices. In this way,

and in each work event must enter time as well must enter time.

The number of all combinations of this kind is equal to i.e. the number of ways that you can from experiences choose in which the event occurred . The probability of each such combination, according to the multiplication theorem for independent events, is equal to . Since combinations are incompatible with each other, then, by the addition theorem, the probability of an event equals

Thus, we can give the following formulation of a particular theorem on the repetition of experiments.

If produced independent experiences in each of which the event appears with probability then the probability that an event will appear exactly times expressed by the formula

, (4.1.1)

Where .

Formula (4.1.1) describes how probabilities are distributed between possible values ​​of a certain random variable — the number of occurrences of the event. at experiences.

Due to the fact that the probabilities according to their form they are members of the binomial decomposition The probability distribution of the form (4.1.1) is called the binomial distribution.