Probability theory. Probability of an event, random events (probability theory)

Classic definition of probability.

The probability of an event is a quantitative measure that is introduced to compare events according to the degree of possibility of their occurrence.

An event that can be represented as a collection (sum) of several elementary events is called composite.

An event that cannot be broken down into simpler ones is called elementary.

An event is called impossible if it never occurs under the conditions of a given experiment (test).

Certain and impossible events are not random.

Joint events– several events are called joint if, as a result of the experiment, the occurrence of one of them does not exclude the occurrence of others.

Incompatible events– several events are called incompatible in a given experiment if the occurrence of one of them excludes the occurrence of others. The two events are called opposite, if one of them occurs if and only if the other does not occur.

The probability of event A is P(A) – is called the number ratio m elementary events (outcomes) favorable to the occurrence of the event A, to the number n all elementary events under the conditions of a given probabilistic experiment.

The following properties of probability follow from the definition:

1. The probability of a random event is a positive number between 0 and 1:

(2)

2. The probability of a certain event is 1: (3)

3. If an event is impossible, then its probability is equal to

(4)

4. If events are incompatible, then

5. If events A and B are joint, then the probability of their sum is equal to the sum of the probabilities of these events without the probability of their joint occurrence:

P(A+B) = P(A) + P(B) - P(AB)(6)

6. If and are opposite events, then (7)

7. Sum of event probabilities A 1, A 2, …, A n, forming a complete group, is equal to 1:

P(A 1) + P(A 2) + …+ P(A n) = 1.(8)

In economic studies, the values ​​and in the formula may be interpreted differently. At statistical definition The probability of an event is the number of observations of the experimental results in which the event occurred exactly once. In this case the relation is called relative frequency (frequency) of an event

Events A, B are called independent, if the probability of each of them does not depend on whether another event occurred or not. The probabilities of independent events are called unconditional.

Events A, B are called dependent, if the probability of each of them depends on whether another event occurred or not. The probability of event B, calculated under the assumption that another event A has already occurred, is called conditional probability.

If two events A and B are independent, then the equalities are true:

P(B) = P(B/A), P(A) = P(A/B) or P(B/A) – P(B) = 0(9)

The probability of the product of two dependent events A, B is equal to the product of the probability of one of them by the conditional probability of the other:

P(AB) = P(B) ∙ P(A/B) or P(AB) = P(A) ∙ P(B/A) (10)

Probability of event B given the occurrence of event A:

(11)

Probability of the product of two independent events A, B is equal to the product of their probabilities:

P(AB) = P(A) ∙ P(B)(12)

If several events are pairwise independent, then it does not follow that they are independent in the aggregate.

Events A 1, A 2, ..., A n (n>2) are called independent in the aggregate if the probability of each of them does not depend on whether any of the other events occurred or not.

The probability of the joint occurrence of several events that are independent in the aggregate is equal to the product of the probabilities of these events:

P(A 1 ∙A 2 ∙A 3 ∙…∙A n) = P(A 1)∙P(A 2)∙P(A 3)∙…∙P(A n). (13)

If, when an event occurs, the probability of the event does not change, then events And are called independent.

Theorem:Probability of co-occurrence of two independent events And (works And ) is equal to the product of the probabilities of these events.

Indeed, since events And are independent, then
. In this case, the formula for the probability of events occurring is And takes on the form.

Events
are called pairwise independent, if any two of them are independent.

Events
are called jointly independent (or simply independent), if every two of them are independent and every event and all possible products of the others are independent.

Theorem:Probability of the product of a finite number of independently independent events
is equal to the product of the probabilities of these events.

Let us illustrate the difference in the application of formulas for the probability of a product of events for dependent and independent events using examples

Example 1. The probability of the first shooter hitting the target is 0.85, the second 0.8. The guns fired one shot each. What is the probability that at least one shell hit the target?

Solution: P(A+B) =P(A) +P(B) –P(AB) Since the shots are independent, then

P(A+B) = P(A) +P(B) –P(A)*P(B) = 0.97

Example 2. The urn contains 2 red and 4 black balls. 2 balls are taken out of it in a row. What is the probability that both balls are red?

Solution: 1 case. Event A is the appearance of a red ball on the first draw, event B on the second. Event C – the appearance of two red balls.

P(C) =P(A)*P(B/A) = (2/6)*(1/5) = 1/15

Case 2. The first ball drawn is returned to the basket

P(C) =P(A)*P(B) = (2/6)*(2/6) = 1/9

Total probability formula.

Let the event can only happen with one of the incompatible events
, forming a complete group. For example, a store receives the same products from three enterprises and in different quantities. The likelihood of producing low-quality products at these enterprises varies. One of the products is randomly selected. It is required to determine the probability that this product is of poor quality (event ). Events here
– this is the selection of a product from the products of the corresponding enterprise.

In this case, the probability of the event can be considered as the sum of the products of events
.

Using the theorem for adding the probabilities of incompatible events, we obtain
. Using the probability multiplication theorem, we find

.

The resulting formula is called total probability formula.

Bayes formula

Let the event occurs simultaneously with one of incompatible events
, the probabilities of which
(
) are known before experiment ( a priori probabilities). An experiment is carried out, as a result of which the occurrence of an event is registered , and it is known that this event had certain conditional probabilities
(
). We need to find the probabilities of events
if it is known that the event happened ( a posteriori probabilities).

The problem is that, having new information (event A occurred), we need to reestimate the probabilities of events
.

Based on the theorem on the probability of the product of two events

.

The resulting formula is called Bayes formulas.

Basic concepts of combinatorics.

When solving a number of theoretical and practical problems, it is necessary to create various combinations from a finite set of elements according to given rules and count the number of all possible such combinations. Such tasks are usually called combinatorial.

When solving problems, combinatorists use the rules of sum and product.

The dependence of events is understood in probabilistic sense, not functional. This means that based on the occurrence of one of the dependent events, one cannot unambiguously judge the occurrence of another. Probabilistic dependence means that the occurrence of one of the dependent events only changes the probability of the occurrence of the other. If the probability does not change, then the events are considered independent.

Definition: Let be an arbitrary probability space, and be some random events. They say that event A does not depend on the event IN , if its conditional probability coincides with the unconditional probability:

.

If , then they say that the event A depends on the event IN.

The concept of independence is symmetrical, that is, if an event A does not depend on the event IN, then the event IN does not depend on the event A. Indeed, let . Then . Therefore they simply say that events A And IN independent.

The following symmetric definition of the independence of events follows from the rule of multiplication of probabilities.

Definition: Events A And IN, defined on the same probability space are called independent, If

If , then events A And IN are called dependent.

Note that this definition is also valid in the case when or .

Properties of independent events.

1. If events A And IN are independent, then the following pairs of events are also independent: .

▲ Let us prove, for example, the independence of events. Let's imagine an event A as: . Since the events are incompatible, then , and due to the independence of the events A And IN we get that . This is what independence means. ■

2. If the event A does not depend on events IN 1 And AT 2, which are inconsistent () , that event A does not depend on the amount.

▲ Indeed, using the axiom of additivity of probability and independence of the event A from events IN 1 And AT 2, we have:

The relationship between the concepts of independence and incompatibility.

Let A And IN- any events that have a non-zero probability: , so . If the events A And IN are inconsistent (), then equality can never take place. Thus, incompatible events are dependent.

When more than two events are considered simultaneously, their pairwise independence does not sufficiently characterize the relationship between the events of the entire group. In this case, the concept of independence in the aggregate is introduced.

Definition: Events defined on the same probability space are called collectively independent, if for any 2 £ m £ n and any combination of indices the equality is true:

At m = 2 From independence in the aggregate follows pairwise independence of events. The reverse is not true.

Example. (Bernstein S.N.)

A random experiment involves tossing a regular tetrahedron (tetrahedron). A face that has fallen down is observed. The faces of the tetrahedron are colored as follows: 1st face - white, 2nd face - black,
The 3rd side is red, the 4th side contains all colors.

Let's consider the events:

A= (white dropout); B= (black dropout);

C= (Red drop).

Then ;

Therefore, events A, IN And WITH are pairwise independent.

However, .

Therefore events A, IN And WITH are not collectively independent.

In practice, as a rule, the independence of events is not established by checking it by definition, but on the contrary: events are considered independent from some external considerations or taking into account the circumstances of a random experiment, and independence is used to find the probabilities of events occurring.

Theorem (multiplication of probabilities for independent events).

If events defined on the same probability space are independent in the aggregate, then the probability of their product is equal to the product of the probabilities:

▲ The proof of the theorem follows from the definition of the independence of events in the aggregate or from the general theorem of multiplication of probabilities, taking into account the fact that in this case

Example 1 (typical example on finding conditional probabilities, the concept of independence, the theorem of addition of probabilities).

The electrical circuit consists of three independently operating elements. The failure probabilities of each element are respectively equal.

1) Find the probability of failure of the circuit.

2) It is known that the circuit has failed.

What is the probability that it refused:

a) 1st element; b) 3rd element?

Solution. Consider the events = (Refused k th element), and event A= (The circuit has failed). Then the event A is presented as:

.

1) Since the events are not incompatible, the axiom of additivity of probability P3) is not applicable and to find the probability one should use the general theorem of addition of probabilities, according to which

General statement of the problem: the probabilities of some events are known, and you need to calculate the probabilities of other events that are associated with these events. In these problems, there is a need for operations with probabilities such as addition and multiplication of probabilities.

For example, while hunting, two shots are fired. Event A- hitting a duck with the first shot, event B- hit from the second shot. Then the sum of events A And B- hit with the first or second shot or with two shots.

Problems of a different type. Several events are given, for example, a coin is tossed three times. You need to find the probability that either the coat of arms will appear all three times, or that the coat of arms will appear at least once. This is a probability multiplication problem.

Addition of probabilities of incompatible events

Addition of probabilities is used when you need to calculate the probability of a combination or logical sum of random events.

Sum of events A And B denote A + B or A ∪ B. The sum of two events is an event that occurs if and only if at least one of the events occurs. It means that A + B– an event that occurs if and only if the event occurred during observation A or event B, or simultaneously A And B.

If events A And B are mutually inconsistent and their probabilities are given, then the probability that one of these events will occur as a result of one trial is calculated using the addition of probabilities.

Probability addition theorem. The probability that one of two mutually incompatible events will occur is equal to the sum of the probabilities of these events:

For example, while hunting, two shots are fired. Event A– hitting a duck with the first shot, event IN– hit from the second shot, event ( A+ IN) – a hit from the first or second shot or from two shots. So, if two events A And IN– incompatible events, then A+ IN– the occurrence of at least one of these events or two events.

Example 1. There are 30 balls of the same size in a box: 10 red, 5 blue and 15 white. Calculate the probability that a colored (not white) ball will be picked up without looking.

Solution. Let us assume that the event A- “the red ball is taken”, and the event IN- “The blue ball was taken.” Then the event is “a colored (not white) ball is taken.” Let's find the probability of the event A:

and events IN:

Events A And IN– mutually incompatible, since if one ball is taken, then it is impossible to take balls of different colors. Therefore, we use the addition of probabilities:

The theorem for adding probabilities for several incompatible events. If events constitute a complete set of events, then the sum of their probabilities is equal to 1:

The sum of the probabilities of opposite events is also equal to 1:

Opposite events form a complete set of events, and the probability of a complete set of events is 1.

Probabilities of opposite events are usually indicated in small letters p And q. In particular,

from which the following formulas for the probability of opposite events follow:

Example 2. The target in the shooting range is divided into 3 zones. The probability that a certain shooter will shoot at the target in the first zone is 0.15, in the second zone – 0.23, in the third zone – 0.17. Find the probability that the shooter will hit the target and the probability that the shooter will miss the target.

Solution: Find the probability that the shooter will hit the target:

Let's find the probability that the shooter will miss the target:

For more complex problems, in which you need to use both addition and multiplication of probabilities, see the page "Various problems involving addition and multiplication of probabilities".

Addition of probabilities of mutually simultaneous events

Two random events are called joint if the occurrence of one event does not exclude the occurrence of a second event in the same observation. For example, when throwing a die the event A The number 4 is considered to be rolled out, and the event IN– rolling an even number. Since 4 is an even number, the two events are compatible. In practice, there are problems of calculating the probabilities of the occurrence of one of the mutually simultaneous events.

Probability addition theorem for joint events. The probability that one of the joint events will occur is equal to the sum of the probabilities of these events, from which the probability of the common occurrence of both events is subtracted, that is, the product of the probabilities. The formula for the probabilities of joint events has the following form:

Since events A And IN compatible, event A+ IN occurs if one of three possible events occurs: or AB. According to the theorem of addition of incompatible events, we calculate as follows:

Event A will occur if one of two incompatible events occurs: or AB. However, the probability of the occurrence of one event from several incompatible events is equal to the sum of the probabilities of all these events:

Likewise:

Substituting expressions (6) and (7) into expression (5), we obtain the probability formula for joint events:

When using formula (8), it should be taken into account that events A And IN can be:

• mutually independent;
• mutually dependent.

Probability formula for mutually independent events:

Probability formula for mutually dependent events:

If events A And IN are inconsistent, then their coincidence is an impossible case and, thus, P(AB) = 0. The fourth probability formula for incompatible events is:

Example 3. In auto racing, when you drive the first car, you have a better chance of winning, and when you drive the second car. Find:

• the probability that both cars will win;
• the probability that at least one car will win;

1) The probability that the first car will win does not depend on the result of the second car, so the events A(the first car wins) and IN(the second car will win) – independent events. Let's find the probability that both cars win:

2) Find the probability that one of the two cars will win:

For more complex problems, in which you need to use both addition and multiplication of probabilities, see the page "Various problems involving addition and multiplication of probabilities".

Solve the addition of probabilities problem yourself, and then look at the solution

Example 4. Two coins are tossed. Event A- loss of the coat of arms on the first coin. Event B- loss of the coat of arms on the second coin. Find the probability of an event C = A + B .

Multiplying Probabilities

Probability multiplication is used when the probability of a logical product of events must be calculated.

In this case, random events must be independent. Two events are said to be mutually independent if the occurrence of one event does not affect the probability of the occurrence of the second event.

Probability multiplication theorem for independent events. Probability of simultaneous occurrence of two independent events A And IN is equal to the product of the probabilities of these events and is calculated by the formula:

Example 5. The coin is tossed three times in a row. Find the probability that the coat of arms will appear all three times.

Solution. The probability that the coat of arms will appear on the first toss of a coin, the second time, and the third time. Let's find the probability that the coat of arms will appear all three times:

Solve probability multiplication problems on your own and then look at the solution

Example 6. There is a box of nine new tennis balls. To play, three balls are taken, and after the game they are put back. When choosing balls, played balls are not distinguished from unplayed balls. What is the probability that after three games there will be no unplayed balls left in the box?

Example 7. 32 letters of the Russian alphabet are written on cut-out alphabet cards. Five cards are drawn at random one after another and placed on the table in order of appearance. Find the probability that the letters will form the word "end".

Example 8. From a full deck of cards (52 sheets), four cards are taken out at once. Find the probability that all four of these cards will be of different suits.

Example 9. The same task as in example 8, but each card after being removed is returned to the deck.

More complex problems, in which you need to use both addition and multiplication of probabilities, as well as calculate the product of several events, can be found on the page "Various problems involving addition and multiplication of probabilities".

The probability that at least one of the mutually independent events will occur can be calculated by subtracting from 1 the product of the probabilities of opposite events, that is, using the formula.

In the Unified State Examination tasks in mathematics, there are also more complex probability problems (than we considered in Part 1), where we have to apply the rule of addition, multiplication of probabilities, and distinguish between compatible and incompatible events.

So, the theory.

Joint and non-joint events

Events are called incompatible if the occurrence of one of them excludes the occurrence of others. That is, only one specific event or another can happen.

For example, when throwing a die, you can distinguish between events such as getting an even number of points and getting an odd number of points. These events are incompatible.

Events are called joint if the occurrence of one of them does not exclude the occurrence of the other.

For example, when throwing a die, you can distinguish such events as rolling an odd number of points and rolling a number of points that are a multiple of three. When a three is rolled, both events occur.

Sum of events

The sum (or combination) of several events is an event consisting of the occurrence of at least one of these events.

Wherein sum of two incompatible events is the sum of the probabilities of these events:

For example, the probability of getting 5 or 6 points on a die with one throw will be , because both events (rolling 5, rolling 6) are inconsistent and the probability of one or the other event occurring is calculated as follows:

The probability sum of two joint events equal to the sum of the probabilities of these events without taking into account their joint occurrence:

For example, in a shopping center, two identical machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Let's find the probability that by the end of the day the coffee will run out in at least one of the machines (that is, either one, or the other, or both at once).

The probability of the first event “coffee will run out in the first machine” as well as the probability of the second event “coffee will run out in the second machine” according to the condition is equal to 0.3. Events are collaborative.

The probability of the joint occurrence of the first two events according to the condition is 0.12.

This means that the probability that by the end of the day the coffee will run out in at least one of the machines is

Dependent and independent events

Two random events A and B are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Otherwise, events A and B are called dependent.

For example, when two dice are rolled simultaneously, one of them, say 1, and the other, 5, are independent events.

Product of probabilities

The product (or intersection) of several events is an event consisting of the joint occurrence of all these events.

If two occur independent events A and B with probabilities P(A) and P(B) respectively, then the probability of the occurrence of events A and B at the same time is equal to the product of the probabilities:

For example, we are interested in seeing a six appear on a die twice in a row. Both events are independent and the probability of each of them occurring separately is . The probability that both of these events will occur will be calculated using the above formula: .

See a selection of tasks to practice the topic.