Probability is one of the basic concepts of probability theory. There are several definitions of this concept. Let us give a definition that is called classical.
Probability event is the ratio of the number of elementary outcomes favorable for a given event to the number of all equally possible outcomes of the experience in which this event may appear.
The probability of event A is denoted by P(A)(Here R– the first letter of a French word probabilite- probability).
According to the definition
where is the number of elementary test outcomes favorable to the occurrence of the event;
The total number of possible elementary test outcomes.
This definition of probability is called classic. It arose at the initial stage of the development of probability theory.
The number is often called the relative frequency of occurrence of an event A in experience.
The greater the probability of an event, the more often it occurs, and vice versa, the less probability of an event, the less often it occurs. When the probability of an event is close to or equal to one, then it occurs in almost all trials. Such an event is said to be almost certain, i.e. that one can certainly count on its occurrence.
On the contrary, when the probability is zero or very small, then the event occurs extremely rarely; such an event is said to be almost impossible.
Sometimes the probability is expressed as a percentage: P(A) 100% is the average percentage of the number of occurrences of an event A.
Example 2.13. While dialing a phone number, the subscriber forgot one digit and dialed it at random. Find the probability that the correct number is dialed.
Solution.
Let us denote by A event - “the required number has been dialed.”
The subscriber could dial any of the 10 digits, so the total number of possible elementary outcomes is 10. These outcomes are incompatible, equally possible and form a complete group. Favors the event A only one outcome (there is only one required number).
The required probability is equal to the ratio of the number of outcomes favorable to the event to the number of all elementary outcomes:
The classical probability formula provides a very simple, experiment-free way to calculate probabilities. However, the simplicity of this formula is very deceptive. The fact is that when using it, two very difficult questions usually arise:
1. How to choose a system of experimental outcomes so that they are equally possible, and is it possible to do this at all?
2. How to find numbers m And n?
If several objects are involved in an experiment, it is not always easy to see equally possible outcomes.
The great French philosopher and mathematician D'Alembert entered the history of probability theory with his famous mistake, the essence of which was that he incorrectly determined the equipossibility of outcomes in an experiment with only two coins!
Example 2.14. ( d'Alembert's error). Two identical coins are tossed. What is the probability that they will fall on the same side?
D'Alembert's solution.
The experiment has three equally possible outcomes:
1. Both coins will land on heads;
2. Both coins will land on tails;
3. One of the coins will land on heads, the other on tails.
The right decision.
The experiment has four equally possible outcomes:
1. The first coin will fall on heads, the second will also fall on heads;
2. The first coin will land on tails, the second will also land on tails;
3. The first coin will fall on heads, and the second on tails;
4. The first coin will land on tails, and the second on heads.
Of these, two outcomes will be favorable for our event, so the required probability is equal to .
D'Alembert made one of the most common mistakes made when calculating probability: he combined two elementary outcomes into one, thereby making it unequal in probability to the remaining outcomes of the experiment.
“Accidents are not accidental”... It sounds like something a philosopher said, but in fact, studying randomness is the destiny of the great science of mathematics. In mathematics, chance is dealt with by probability theory. Formulas and examples of tasks, as well as the basic definitions of this science will be presented in the article.
What is probability theory?
Probability theory is one of the mathematical disciplines that studies random events.
To make it a little clearer, let's give a small example: if you throw a coin up, it can land on heads or tails. While the coin is in the air, both of these probabilities are possible. That is, the probability of possible consequences is 1:1. If one is drawn from a deck of 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict here, especially with the help of mathematical formulas. However, if you repeat a certain action many times, you can identify a certain pattern and, based on it, predict the outcome of events in other conditions.
To summarize all of the above, probability theory in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical value.
From the pages of history
The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.
Initially, probability theory had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. They studied gambling for a long time and saw certain patterns, which they decided to tell the public about.
The same technique was invented by Christiaan Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of “probability theory”, formulas and examples, which are considered the first in the history of the discipline, were introduced by him.
The works of Jacob Bernoulli, Laplace's and Poisson's theorems are also of no small importance. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks received their current form thanks to Kolmogorov’s axioms. As a result of all the changes, probability theory became one of the mathematical branches.
Basic concepts of probability theory. Events
The main concept of this discipline is “event”. There are three types of events:
- Reliable. Those that will happen anyway (the coin will fall).
- Impossible. Events that will not happen under any circumstances (the coin will remain hanging in the air).
- Random. The ones that will happen or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then there are random factors that can affect the result: the physical characteristics of the coin, its shape, its original position, the force of the throw, etc.
All events in the examples are indicated by capital Latin letters, with the exception of P, which has a different role. For example:
- A = “students came to lecture.”
- Ā = “students did not come to the lecture.”
In practical tasks, events are usually written down in words.
One of the most important characteristics of events is their equal possibility. That is, if you toss a coin, all options for the initial fall are possible until it falls. But events are also not equally possible. This happens when someone deliberately influences an outcome. For example, “marked” playing cards or dice, in which the center of gravity is shifted.
Events can also be compatible and incompatible. Compatible events do not exclude each other's occurrence. For example:
- A = “the student came to the lecture.”
- B = “the student came to the lecture.”
These events are independent of each other, and the occurrence of one of them does not affect the occurrence of the other. Incompatible events are defined by the fact that the occurrence of one excludes the occurrence of another. If we talk about the same coin, then the loss of “tails” makes it impossible for the appearance of “heads” in the same experiment.
Actions on events
Events can be multiplied and added; accordingly, logical connectives “AND” and “OR” are introduced in the discipline.
The amount is determined by the fact that either event A or B, or two, can occur at the same time. If they are incompatible, the last option is impossible; either A or B will be rolled.
Multiplication of events consists in the appearance of A and B at the same time.
Now we can give several examples to better remember the basics, probability theory and formulas. Examples of problem solving below.
Task 1: The company takes part in a competition to receive contracts for three types of work. Possible events that may occur:
- A = “the firm will receive the first contract.”
- A 1 = “the firm will not receive the first contract.”
- B = “the firm will receive a second contract.”
- B 1 = “the firm will not receive a second contract”
- C = “the firm will receive a third contract.”
- C 1 = “the firm will not receive a third contract.”
Using actions on events, we will try to express the following situations:
- K = “the company will receive all contracts.”
In mathematical form, the equation will have the following form: K = ABC.
- M = “the company will not receive a single contract.”
M = A 1 B 1 C 1.
Let’s complicate the task: H = “the company will receive one contract.” Since it is not known which contract the company will receive (first, second or third), it is necessary to record the entire series of possible events:
H = A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.
And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second. Other possible events were recorded using the appropriate method. The symbol υ in the discipline denotes the connective “OR”. If we translate the above example into human language, the company will receive either the third contract, or the second, or the first. In a similar way, you can write down other conditions in the discipline “Probability Theory”. The formulas and examples of problem solving presented above will help you do this yourself.
Actually, the probability
Perhaps, in this mathematical discipline, the probability of an event is the central concept. There are 3 definitions of probability:
- classic;
- statistical;
- geometric.
Each has its place in the study of probability. Probability theory, formulas and examples (9th grade) mainly use the classic definition, which sounds like this:
- The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.
The formula looks like this: P(A)=m/n.
A is actually an event. If a case opposite to A appears, it can be written as Ā or A 1 .
m is the number of possible favorable cases.
n - all events that can happen.
For example, A = “draw a card of the heart suit.” There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:
P(A)=9/36=0.25.
As a result, the probability that a card of the heart suit will be drawn from the deck will be 0.25.
Towards higher mathematics
Now it has become a little known what the theory of probability is, formulas and examples of solving problems that come across in the school curriculum. However, probability theory is also found in higher mathematics, which is taught in universities. Most often they operate with geometric and statistical definitions of the theory and complex formulas.
The theory of probability is very interesting. It is better to start studying formulas and examples (higher mathematics) small - with the statistical (or frequency) definition of probability.
The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic one:
If the classical formula is calculated for prediction, then the statistical one is calculated according to the results of the experiment. Let's take a small task for example.
The technological control department checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?
A = “the appearance of a quality product.”
W n (A)=97/100=0.97
Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Out of 100 products that were checked, 3 were found to be of poor quality. We subtract 3 from 100, we get 97, this is the amount of quality goods.
A little about combinatorics
Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B can be made in n different ways, then the choice of A and B can be made by multiplication.
For example, there are 5 roads leading from city A to city B. There are 4 paths from city B to city C. In how many ways can you get from city A to city C?
It's simple: 5x4=20, that is, in twenty different ways you can get from point A to point C.
Let's complicate the task. How many ways are there to lay out cards in solitaire? There are 36 cards in the deck - this is the starting point. To find out the number of ways, you need to “subtract” one card at a time from the starting point and multiply.
That is, 36x35x34x33x32...x2x1= the result does not fit on the calculator screen, so it can simply be designated 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied together.
In combinatorics there are such concepts as permutation, placement and combination. Each of them has its own formula.
An ordered set of elements of a set is called an arrangement. Placements can be repeated, that is, one element can be used several times. And without repetition, when elements are not repeated. n are all elements, m are elements that participate in the placement. The formula for placement without repetition will look like:
A n m =n!/(n-m)!
Connections of n elements that differ only in the order of placement are called permutations. In mathematics it looks like: P n = n!
Combinations of n elements of m are those compounds in which it is important what elements they were and what their total number is. The formula will look like:
A n m =n!/m!(n-m)!
Bernoulli's formula
In probability theory, as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the occurrence of A in an experiment does not depend on the occurrence or non-occurrence of the same event in earlier or subsequent trials.
Bernoulli's equation:
P n (m) = C n m ×p m ×q n-m.
The probability (p) of the occurrence of event (A) is constant for each trial. The probability that the situation will occur exactly m times in n number of experiments will be calculated by the formula presented above. Accordingly, the question arises of how to find out the number q.
If event A occurs p number of times, accordingly, it may not occur. Unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that denotes the possibility of an event not occurring.
Now you know Bernoulli's formula (probability theory). We will consider examples of problem solving (first level) below.
Task 2: A store visitor will make a purchase with probability 0.2. 6 visitors independently entered the store. What is the likelihood that a visitor will make a purchase?
Solution: Since it is unknown how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.
A = “the visitor will make a purchase.”
In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.
n = 6 (since there are 6 customers in the store). The number m will vary from 0 (not a single customer will make a purchase) to 6 (all visitors to the store will purchase something). As a result, we get the solution:
P 6 (0) = C 0 6 ×p 0 ×q 6 =q 6 = (0.8) 6 = 0.2621.
None of the buyers will make a purchase with probability 0.2621.
How else is Bernoulli's formula (probability theory) used? Examples of problem solving (second level) below.
After the above example, questions arise about where C and r went. Relative to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:
C n m = n! /m!(n-m)!
Since in the first example m = 0, respectively, C = 1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of two visitors purchasing goods.
P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.
The theory of probability is not that complicated. Bernoulli's formula, examples of which are presented above, is direct proof of this.
Poisson's formula
Poisson's equation is used to calculate low-probability random situations.
Basic formula:
P n (m)=λ m /m! × e (-λ) .
In this case λ = n x p. Here is a simple Poisson formula (probability theory). We will consider examples of problem solving below.
Task 3: The factory produced 100,000 parts. Occurrence of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?
As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of solving problems of this kind are no different from other tasks in the discipline; we substitute the necessary data into the given formula:
A = “a randomly selected part will be defective.”
p = 0.0001 (according to the task conditions).
n = 100000 (number of parts).
m = 5 (defective parts). We substitute the data into the formula and get:
R 100000 (5) = 10 5 /5! X e -10 = 0.0375.
Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In fact, it can be found by the formula:
e -λ = lim n ->∞ (1-λ/n) n .
However, there are special tables that contain almost all values of e.
De Moivre-Laplace theorem
If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of occurrence of event A in all schemes is the same, then the probability of occurrence of event A a certain number of times in a series of tests can be found by Laplace’s formula:
Р n (m)= 1/√npq x ϕ(X m).
X m = m-np/√npq.
To better remember Laplace’s formula (probability theory), examples of problems are below to help.
First, let's find X m, substitute the data (they are all listed above) into the formula and get 0.025. Using tables, we find the number ϕ(0.025), the value of which is 0.3988. Now you can substitute all the data into the formula:
P 800 (267) = 1/√(800 x 1/3 x 2/3) x 0.3988 = 3/40 x 0.3988 = 0.03.
Thus, the probability that the flyer will work exactly 267 times is 0.03.
Bayes formula
The Bayes formula (probability theory), examples of solving problems with the help of which will be given below, is an equation that describes the probability of an event based on the circumstances that could be associated with it. The basic formula is as follows:
P (A|B) = P (B|A) x P (A) / P (B).
A and B are definite events.
P(A|B) is a conditional probability, that is, event A can occur provided that event B is true.
P (B|A) - conditional probability of event B.
So, the final part of the short course “Theory of Probability” is the Bayes formula, examples of solutions to problems with which are below.
Task 5: Phones from three companies were brought to the warehouse. At the same time, the share of phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. You need to find the probability that a randomly selected phone will be defective.
A = “randomly picked phone.”
B 1 - the phone that the first factory produced. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).
As a result we get:
P (B 1) = 25%/100% = 0.25; P(B 2) = 0.6; P (B 3) = 0.15 - thus we found the probability of each option.
Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in companies:
P (A/B 1) = 2%/100% = 0.02;
P(A/B 2) = 0.04;
P (A/B 3) = 0.01.
Now let’s substitute the data into the Bayes formula and get:
P (A) = 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 = 0.0305.
The article presents probability theory, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after everything that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It’s difficult for an ordinary person to answer; it’s better to ask someone who has used it to win the jackpot more than once.
Basics of probability theory
Plan:
1. Random events
2. Classic definition of probability
3. Calculation of event probabilities and combinatorics
4. Geometric probability
Theoretical information
Random events.
Random phenomenon- a phenomenon whose outcome is not clearly defined. This concept can be interpreted in a fairly broad sense. Namely: everything in nature is quite random, the appearance and birth of any individual is a random phenomenon, choosing a product in a store is also a random phenomenon, getting a grade on an exam is a random phenomenon, illness and recovery are random phenomena, etc.
Examples of random phenomena:
~ Firing is carried out from a gun mounted at a given angle to the horizontal. Hitting the target is accidental, but the projectile hitting a certain “fork” is a pattern. You can specify the distance closer to which and further than which the projectile will not fly. You will get some kind of “projectile dispersion fork”
~ The same body is weighed several times. Strictly speaking, each time you will get different results, even if they differ by an insignificant amount, but they will be different.
~ An airplane, flying along the same route, has a certain flight corridor within which the airplane can maneuver, but it will never have a strictly identical route
~ An athlete will never be able to run the same distance in the same time. Its results will also be within a certain numerical range.
Experience, experiment, observation are tests
Trial– observation or fulfillment of a certain set of conditions that are fulfilled repeatedly, and regularly repeated in the same sequence, duration, and in compliance with other identical parameters.
Let's consider an athlete firing at a target. In order for it to be carried out, it is necessary to fulfill such conditions as preparing the athlete, loading the weapon, aiming, etc. “Hit” and “missed” – events as a result of a shot.
Event– high-quality test result.
An event may or may not happen. Events are indicated in capital letters. For example: D = "The shooter hit the target." S="The white ball is drawn." K="A lottery ticket taken at random without winning.".
Tossing a coin is a test. The fall of her “coat of arms” is one event, the fall of her “digit” is the second event.
Any test involves the occurrence of several events. Some of them may be necessary for the researcher at a given time, others may not be necessary.
The event is called random, if when a certain set of conditions is met S it can either happen or not happen. In what follows, instead of saying “the set of conditions S has been fulfilled,” we will say briefly: “the test has been carried out.” Thus, the event will be considered as the result of the test.
~ The shooter shoots at a target divided into four areas. The shot is a test. Hitting a certain area of the target is an event.
~ There are colored balls in the urn. One ball is taken at random from the urn. Retrieving a ball from an urn is a test. The appearance of a ball of a certain color is an event.
Types of random events
1. Events are called incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.
~ A part is randomly removed from a parts box. The appearance of a standard part eliminates the appearance of a non-standard part. Events € a standard part appeared" and a non-standard part appeared" - incompatible.
~ A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events “a coat of arms appeared” and “an inscription appeared” are incompatible.
Several events form full group, if at least one of them appears as a result of the test. In other words, the occurrence of at least one of the events of the complete group is a reliable event.
In particular, if the events that form the complete group are pairwise incompatible, then the result of the test will be one and only one of these events. This special case is of greatest interest to us, since it will be used further.
~ Two cash and clothing lottery tickets were purchased. One and only one of the following events is sure to occur:
1. “the winnings fell on the first ticket and did not fall on the second,”
2. “the winnings did not fall on the first ticket and fell on the second,”
3. “the winnings fell on both tickets”,
4. “both tickets did not win.”
These events form a complete group of pairwise incompatible events,
~ The shooter fired at the target. One of the following two events will definitely happen: hit, miss. These two incompatible events also form a complete group.
2. Events are called equally possible, if there is reason to believe that neither of them is more possible than the other.
~ The appearance of a “coat of arms” and the appearance of an inscription when throwing a coin are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of minting does not affect the loss of one side or another of the coin.
~ The appearance of one or another number of points on a thrown dice are equally possible events. Indeed, it is assumed that the die is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of points does not affect the loss of any face.
3. The event is called reliable, if it can't help but happen
4. The event is called unreliable, if it cannot happen.
5. The event is called opposite to some event if it consists of the non-occurrence of this event. Opposite events are not compatible, but one of them must necessarily happen. Opposite events are usually designated as negations, i.e. A dash is written above the letter. Opposite events: A and Ā; U and Ū, etc. .
Classic definition of probability
Probability is one of the basic concepts of probability theory.
There are several definitions of this concept. Let us give a definition that is called classical. Next, we will indicate the weaknesses of this definition and give other definitions that allow us to overcome the shortcomings of the classical definition.
Consider the situation: A box contains 6 identical balls, 2 are red, 3 are blue and 1 is white. Obviously, the possibility of drawing a colored (i.e., red or blue) ball from an urn at random is greater than the possibility of drawing a white ball. This possibility can be characterized by a number, which is called the probability of an event (the appearance of a colored ball).
Probability- a number characterizing the degree of possibility of an event occurring.
In the situation under consideration, we denote:
Event A = "Pulling out a colored ball."
Let us call each of the possible results of the test (the test consists of removing a ball from an urn) elementary (possible) outcome and event. Elementary outcomes can be denoted by letters with indices below, for example: k 1, k 2.
In our example there are 6 balls, so there are 6 possible outcomes: a white ball appears; a red ball appeared; a blue ball appeared, etc. It is easy to see that these outcomes form a complete group of pairwise incompatible events (only one ball will appear) and they are equally possible (the ball is drawn at random, the balls are identical and thoroughly mixed).
Let us call elementary outcomes in which the event of interest to us occurs favorable outcomes this event. In our example, the event is favored A(the appearance of a colored ball) the following 5 outcomes:
So the event A is observed if one of the elementary outcomes favorable to the test occurs, no matter which one. A. This is the appearance of any colored ball, of which there are 5 in the box
In the example under consideration, there are 6 elementary outcomes; 5 of them favor the event A. Hence, P(A)= 5/6. This number gives a quantitative assessment of the degree of possibility of the appearance of a colored ball.
Definition of probability:
Probability of event A is the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form the complete group.
P(A)=m/n or P(A)=m: n, where:
m is the number of elementary outcomes favorable A;
n- the number of all possible elementary test outcomes.
Here it is assumed that the elementary outcomes are incompatible, equally possible and form a complete group.
The following properties follow from the definition of probability:
1. The probability of a reliable event is equal to one.
Indeed, if the event is reliable, then every elementary outcome of the test favors the event. In this case m = n therefore p=1
2. The probability of an impossible event is zero.
Indeed, if an event is impossible, then none of the elementary outcomes of the test favor the event. In this case m=0, therefore p=0.
3.The probability of a random event is a positive number between zero and one. 0
In subsequent topics, theorems will be given that allow one to find the probabilities of other events using the known probabilities of some events.
Measurement. There are 6 girls and 4 boys in the group of students. What is the probability that a randomly selected student will be a girl? will there be a young man?
p dev = 6 / 10 =0.6 p yun = 4 / 10 = 0.4
The concept of “probability” in modern rigorous probability theory courses is built on a set-theoretic basis. Let's look at some aspects of this approach.
Let one and only one of the events occur as a result of the test: w i(i=1, 2, .... p). Events w i- called elementary events (elementary outcomes). ABOUT It follows from here that elementary events are pairwise incompatible. The set of all elementary events that can occur in a test is called space of elementary eventsΩ (capital Greek letter omega), and the elementary events themselves are points of this space..
Event A identified with a subset (of space Ω), the elements of which are elementary outcomes favorable A; event IN is a subset Ω whose elements are outcomes favorable IN, etc. Thus, the set of all events that can occur in a test is the set of all subsets of Ω. Ω itself occurs for any outcome of the test, therefore Ω is a reliable event; an empty subset of space Ω - is an impossible event (it does not occur under any outcome of the test).
Elementary events are distinguished from among all topic events, “each of them contains only one element Ω
Every elementary outcome w i match a positive number p i- the probability of this outcome, and the sum of all p i equal to 1 or with a sum sign, this fact will be written in the form of an expression:
By definition, probability P(A) events A equal to the sum of the probabilities of elementary outcomes favorable A. Therefore, the probability of a reliable event is equal to one, an impossible event is zero, and an arbitrary event is between zero and one.
Let's consider an important special case when all outcomes are equally possible. The number of outcomes is n, the sum of the probabilities of all outcomes is equal to one; therefore, the probability of each outcome is 1/p. Let the event A favors m outcomes.
Probability of event A equal to the sum of the probabilities of outcomes favorable A:
P(A)=1/n + 1/n+…+1/n = n 1/n=1
A classical definition of probability is obtained.
There is also axiomatic approach to the concept of "probability". In the system of axioms proposed. Kolmogorov A.N., undefined concepts are an elementary event and probability. The construction of a logically complete theory of probability is based on the axiomatic definition of a random event and its probability.
Here are the axioms that define probability:
1. Every event A assigned a non-negative real number R(A). This number is called the probability of the event A.
2. The probability of a reliable event is equal to one:
3. The probability of the occurrence of at least one of the pairwise incompatible events is equal to the sum of the probabilities of these events.
Based on these axioms, the properties of probabilities and the dependence between them are derived as theorems.
For practical activities, it is important to be able to compare events according to the degree of possibility of their occurrence. Obviously, the events - “rain” and “snow” on the first day of summer in a given area, “winning on one ticket” and “winning on each of 5 purchased tickets” of a cash and clothing lottery have varying degrees of possibility of their occurrence. Therefore, to compare events, a certain measure is needed.
To quantify the degree of possibility of a random event occurring, the term probability is used.
Let us set the task to give a quantitative estimate of the possibility that when throwing a dice, 4 points will appear. We will consider the loss of four points as event A. Each of the possible results of the test (test - throwing a die) will be called an elementary outcome (elementary event). In our example, the following 6 elementary outcomes are possible: 1 point, 2 points, 3 points, 4 points, 5 points, 6 points. We will call those elementary outcomes in which the event of interest to us occurs favorable to this event. In our example, out of six elementary outcomes, event A is favored by one. Therefore, the probability that the number of points drawn will be equal to 4 is equal to 1/6. This number gives the quantitative assessment of the degree of possibility of the appearance of four points, which we wanted to find.
According to the classical definition, the probability of event A is equal to the ratio of the number of outcomes favorable to this event to the total number of equally possible elementary outcomes.
The following properties follow from the definition of probability:
Condition 1. The probability of a reliable event is equal to one.
P(A) = t/p = p/p = 1.
Statement 2. The probability of an impossible event is zero.
P(A) = t/p = 0/p = 0.
Property 3. The probability of a random event is a positive number between zero and one.
0
P(A) 
1.
Example 1. A water supply failure occurred on the territory of an enterprise. The total length of the water pipeline is 150 m. Including 50 m of pipe in hard-to-reach places. What is the likelihood that repairs will have to be made in a hard-to-reach area?
P(A) = 50/150 = 1/3
Example 2. An urn contains m white balls and n black balls. What is the probability of drawing a white ball (event A)?
3. Statistical determination of probability.
Using the classical definition of probability, you can calculate the probability of any random event without resorting to experience. However, this is not always feasible, because in practice it is not always possible to comply with the condition of equal opportunity that underlies the classical definition.
For example, if the coin is flattened, then the events “appearance of a coat of arms” and “appearance of a figure” cannot be considered equally possible and formula (1) will be inapplicable for calculating the probability of any of them. For this reason, along with the classical definition, the statistical definition of probability is used.
When studying mass phenomena, a random event or random variable may appear several times during testing. Let, for example, in n trials the event A appears m times. The number m is called the frequency of occurrence of event A. The ratio of the frequency of event A to the total number of trials n is called the event frequency or relative frequency, which is designated
If a random event has a stable frequency in a series of tests, i.e. In each series of tests, the frequency of this event changes slightly and fluctuates around a certain positive number, then this number is taken as the probability of this event. The probability calculated in this way is called statistical probability.
(2)
Example 1. Let's toss a coin 10 times and get, for example, the following results:
|
G, G, C, G, C, |
G, C, G, C, 10) C, |
As the number of tests increases, the frequency fluctuations decrease and the frequency becomes almost stable. Such a stable frequency is assumed to be equal to the probability of the event of interest to us.
In the coin toss example, the number of experiments is taken arbitrarily. In fact, to obtain a reliable probability value, the number of experiments must be much larger.
In economics, as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the sales volume of a product depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and carrying out sales, you have to predict the outcome of such activities on the basis of either your own previous experience, or similar experience of other people, or intuition, which to a large extent also relies on experimental data.
In order to somehow evaluate the event in question, it is necessary to take into account or specially organize the conditions in which this event is recorded.
The implementation of certain conditions or actions to identify the event in question is called experience or experiment.
The event is called random, if as a result of experience it may or may not occur.
The event is called reliable, if it necessarily appears as a result of a given experience, and impossible, if it cannot appear in this experience.
For example, snowfall in Moscow on November 30 is a random event. The daily sunrise can be considered a reliable event. Snowfall at the equator can be considered an impossible event.
One of the main tasks in probability theory is the task of determining a quantitative measure of the possibility of an event occurring.
Algebra of events
Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.
Amount events is an event consisting of the occurrence of at least one of these events
An example of the sum of events is the presence of at least one of two products in the store.
The work events is an event consisting of the simultaneous occurrence of all these events
An event consisting of the appearance of two goods in a store at the same time is a product of events: - the appearance of one product, - the appearance of another product.
Events form a complete group of events if at least one of them is sure to occur in experience.
Example. The port has two berths for receiving ships. Three events can be considered: - the absence of ships at the berths, - the presence of one ship at one of the berths, - the presence of two ships at two berths. These three events form a complete group of events.
Opposite two unique possible events that form a complete group are called.
If one of the events that is opposite is denoted by , then the opposite event is usually denoted by .
Classical and statistical definitions of event probability
Each of the equally possible results of tests (experiments) is called an elementary outcome. They are usually designated by letters. For example, a die is thrown. There can be a total of six elementary outcomes based on the number of points on the sides.
From elementary outcomes you can create a more complex event. Thus, the event of an even number of points is determined by three outcomes: 2, 4, 6.
A quantitative measure of the possibility of the occurrence of the event in question is probability.
The most widely used definitions of the probability of an event are: classic And statistical.
The classical definition of probability is associated with the concept of a favorable outcome.
The outcome is called favorable to a given event if its occurrence entails the occurrence of this event.
In the above example, the event in question—an even number of points on the rolled side—has three favorable outcomes. In this case, the general
number of possible outcomes. This means that the classical definition of the probability of an event can be used here.
Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes
where is the probability of the event, is the number of outcomes favorable to the event, is the total number of possible outcomes.
In the considered example
The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.
The relative frequency of occurrence of an event is calculated using the formula
where is the number of occurrences of an event in a series of experiments (tests).
Statistical definition. The probability of an event is the number around which the relative frequency stabilizes (sets) with an unlimited increase in the number of experiments.
In practical problems, the probability of an event is taken to be the relative frequency for a sufficiently large number of trials.
From these definitions of the probability of an event it is clear that the inequality is always satisfied
To determine the probability of an event based on formula (1.1), combinatorics formulas are often used, which are used to find the number of favorable outcomes and the total number of possible outcomes.
