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4.1. Private theorem on repetition of experiments

Lecture



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.   4.1.  Private theorem on repetition of experiments , and we are not interested in the result of each individual experience, but in the total number of occurrences   4.1.  Private theorem on repetition of experiments 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   4.1.  Private theorem on repetition of experiments 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   4.1.  Private theorem on repetition of experiments . Find the probability that with these three shots we get exactly two hits.

Decision. Denote   4.1.  Private theorem on repetition of experiments 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   4.1.  Private theorem on repetition of experiments can be represented as the sum of the works of events:

  4.1.  Private theorem on repetition of experiments ,

Where   4.1.  Private theorem on repetition of experiments - hits at the first, second, third shots respectively,   4.1.  Private theorem on repetition of experiments - a miss at the first, second, third shots.

Considering that the three listed event options   4.1.  Private theorem on repetition of experiments incompatible, and the events included in the works are independent, according to the theorems of addition and multiplication we get:

  4.1.  Private theorem on repetition of experiments ,

or denoting   4.1.  Private theorem on repetition of experiments ,

  4.1.  Private theorem on repetition of experiments .

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   4.1.  Private theorem on repetition of experiments independent experiments, in each of which some event may or may not appear   4.1.  Private theorem on repetition of experiments ; probability of occurrence   4.1.  Private theorem on repetition of experiments in each experience is equal   4.1.  Private theorem on repetition of experiments and the probability of non-occurrence   4.1.  Private theorem on repetition of experiments . Required to find the probability   4.1.  Private theorem on repetition of experiments that event   4.1.  Private theorem on repetition of experiments in these   4.1.  Private theorem on repetition of experiments experiences will appear exactly   4.1.  Private theorem on repetition of experiments time.

Consider an event   4.1.  Private theorem on repetition of experiments that event   4.1.  Private theorem on repetition of experiments will appear in   4.1.  Private theorem on repetition of experiments experiences exactly   4.1.  Private theorem on repetition of experiments time. This event can be implemented in various ways. Decompose event   4.1.  Private theorem on repetition of experiments for the sum of the products of events consisting in the occurrence or non-occurrence   4.1.  Private theorem on repetition of experiments in a separate experience. We will denote   4.1.  Private theorem on repetition of experiments occurrence of an event   4.1.  Private theorem on repetition of experiments in the i-th experience;   4.1.  Private theorem on repetition of experiments - non-event   4.1.  Private theorem on repetition of experiments in the i-th experience.

Obviously, each version of the event   4.1.  Private theorem on repetition of experiments (each member of the sum) must consist of m occurrences   4.1.  Private theorem on repetition of experiments and   4.1.  Private theorem on repetition of experiments non-appearances, i.e. of   4.1.  Private theorem on repetition of experiments events   4.1.  Private theorem on repetition of experiments and   4.1.  Private theorem on repetition of experiments events   4.1.  Private theorem on repetition of experiments with different indices. In this way,

  4.1.  Private theorem on repetition of experiments

and in each work event   4.1.  Private theorem on repetition of experiments must enter   4.1.  Private theorem on repetition of experiments time as well   4.1.  Private theorem on repetition of experiments must enter   4.1.  Private theorem on repetition of experiments time.

The number of all combinations of this kind is equal to   4.1.  Private theorem on repetition of experiments i.e. the number of ways that you can from   4.1.  Private theorem on repetition of experiments experiences choose   4.1.  Private theorem on repetition of experiments in which the event occurred   4.1.  Private theorem on repetition of experiments . The probability of each such combination, according to the multiplication theorem for independent events, is equal to   4.1.  Private theorem on repetition of experiments . Since combinations are incompatible with each other, then, by the addition theorem, the probability of an event   4.1.  Private theorem on repetition of experiments equals

  4.1.  Private theorem on repetition of experiments

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

If produced   4.1.  Private theorem on repetition of experiments independent experiences in each of which the event   4.1.  Private theorem on repetition of experiments appears with probability   4.1.  Private theorem on repetition of experiments then the probability that an event   4.1.  Private theorem on repetition of experiments will appear exactly   4.1.  Private theorem on repetition of experiments times expressed by the formula

  4.1.  Private theorem on repetition of experiments , (4.1.1)

Where   4.1.  Private theorem on repetition of experiments .

Formula (4.1.1) describes how probabilities are distributed between possible values ​​of a certain random variable — the number of occurrences of the event.   4.1.  Private theorem on repetition of experiments at   4.1.  Private theorem on repetition of experiments experiences.

Due to the fact that the probabilities   4.1.  Private theorem on repetition of experiments according to their form they are members of the binomial decomposition   4.1.  Private theorem on repetition of experiments The probability distribution of the form (4.1.1) is called the binomial distribution.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis