What is an event in probability theory. Problems on the classical determination of probability. Examples of solutions. Relationships between events
For practical activities, it is necessary to be able to compare events according to the degree of possibility of their occurrence. Let's consider a classic case. There are 10 balls in the urn, 8 of them are white, 2 are black. Obviously, the event “a white ball will be drawn from the urn” and the event “a black ball will be drawn from the urn” have different degrees of possibility of their occurrence. Therefore, to compare events, a certain quantitative measure is needed.
A quantitative measure of the possibility of an event occurring is probability . The most widely used definitions of the probability of an event are classical and statistical.
Classic definition probability is associated with the concept of a favorable outcome. Let's look at this in more detail.
Let the outcomes of some test form a complete group of events and are equally possible, i.e. uniquely possible, incompatible and equally possible. Such outcomes are called elementary outcomes, or cases. It is said that the test boils down to case scheme or " urn scheme", because Any probability problem for such a test can be replaced by an equivalent problem with urns and balls of different colors.
The outcome is called favorable event A, if the occurrence of this case entails the occurrence of the event A.
According to the classical definition probability of an event A is equal to the ratio of the number of outcomes favorable to this event to the total number of outcomes, i.e.
| , | (1.1) |
Where P(A)– probability of event A; m– number of cases favorable to the event A; n– total number of cases.
Example 1.1. When throwing a dice, there are six possible outcomes: 1, 2, 3, 4, 5, 6 points. What is the probability of getting an even number of points?
Solution. All n= 6 outcomes form a complete group of events and are equally possible, i.e. uniquely possible, incompatible and equally possible. Event A - “the appearance of an even number of points” - is favored by 3 outcomes (cases) - the loss of 2, 4 or 6 points. Using the classical formula for the probability of an event, we obtain
P(A) = = . ◄
Based on the classical definition of the probability of an event, we note its properties:
1. The probability of any event lies between zero and one, i.e.
0 ≤ R(A) ≤ 1.
2. The probability of a reliable event is equal to one.
3. The probability of an impossible event is zero.
As stated earlier, the classical definition of probability is applicable only for those events that can arise as a result of tests that have symmetry of possible outcomes, i.e. reducible to a pattern of cases. However, there is a large class of events whose probabilities cannot be calculated using the classical definition.
For example, if we assume that the coin is flattened, then it is obvious that the events “appearance of a coat of arms” and “appearance of heads” cannot be considered equally possible. Therefore, the formula for determining the probability according to the classical scheme is not applicable in this case.
However, there is another approach to estimating the probability of events, based on how often a given event will occur in the trials performed. In this case, the statistical definition of probability is used.
Statistical probabilityevent A is the relative frequency (frequency) of occurrence of this event in n trials performed, i.e.
 ,
| (1.2) |
Where P*(A)– statistical probability of an event A; w(A)– relative frequency of the event A; m– number of trials in which the event occurred A; n– total number of tests.
Unlike mathematical probability P(A), considered in the classical definition, statistical probability P*(A) is a characteristic experienced, experimental. In other words, the statistical probability of an event A is the number around which the relative frequency is stabilized (set) w(A) with an unlimited increase in the number of tests carried out under the same set of conditions.
For example, when they say about a shooter that he hits the target with a probability of 0.95, this means that out of hundreds of shots fired by him under certain conditions (the same target at the same distance, the same rifle, etc. .), on average there are about 95 successful ones. Naturally, not every hundred will have 95 successful shots, sometimes there will be fewer, sometimes more, but on average, with multiple repetitions of shooting under the same conditions, this percentage of hits will remain unchanged. The figure of 0.95, which serves as an indicator of the shooter's skill, is usually very stable, i.e. the percentage of hits in most shootings will be almost the same for a given shooter, only in rare cases deviating any significantly from its average value.
Another disadvantage of the classical definition of probability ( 1.1 ) limiting its use is that it assumes a finite number of possible test outcomes. In some cases, this disadvantage can be overcome by using a geometric definition of probability, i.e. finding the probability of a point falling into a certain area (segment, part of a plane, etc.).
Let the flat figure g forms part of a flat figure G(Fig. 1.1). Fit G a dot is thrown at random. This means that all points in the region G“equal rights” with respect to whether a thrown random point hits it. Assuming that the probability of an event A– the thrown point hits the figure g– is proportional to the area of this figure and does not depend on its location relative to G, neither from the form g, we'll find
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.
The probability of an event is understood as a certain numerical characteristic of the possibility of the occurrence of this event. There are several approaches to determining probability.
Probability of the event A is called 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. So, the probability of the event A is determined by the formula
Where m– the number of elementary outcomes favorable A, n– the number of all possible elementary test outcomes.
Example 3.1. In an experiment involving throwing a die, the number of all outcomes n equals 6 and they are all equally possible. Let the event A means the appearance of an even number. Then for this event, favorable outcomes will be the appearance of numbers 2, 4, 6. Their number is 3. Therefore, the probability of the event A equal to
Example 3.2. What is the probability that a two-digit number chosen at random has the same digits?
Two-digit numbers are numbers from 10 to 99, there are 90 such numbers in total. 9 numbers have identical digits (these are numbers 11, 22, ..., 99). Since in this case m=9, n=90, then
Where A– event, “a number with the same digits.”
Example 3.3. In a batch of 10 parts, 7 are standard. Find the probability that among six parts taken at random, 4 are standard.
The total number of possible elementary test outcomes is equal to the number of ways in which 6 parts can be extracted from 10, i.e., the number of combinations of 10 elements of 6 elements each. Let us determine the number of outcomes favorable to the event of interest to us A(among the six taken parts there are 4 standard ones). Four standard parts can be taken from seven standard parts in different ways; at the same time, the remaining 6-4=2 parts must be non-standard, but you can take two non-standard parts from 10-7=3 non-standard parts in different ways. Therefore, the number of favorable outcomes is equal to .
Then the required probability is equal to
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
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 it means
3. The probability of a random event is a positive number between zero and one.
Indeed, only a part of the total number of elementary outcomes of the test is favored by a random event. In this case< m< n, means 0 < m/n < 1, i.e. 0< P(A) < 1. Итак, вероятность любого события удовлетворяет двойному неравенству
The construction of a logically complete theory of probability is based on the axiomatic definition of a random event and its probability. In the system of axioms proposed by A. N. Kolmogorov, the undefined concepts are an elementary event and probability. Here are the axioms that define probability:
1. Every event A assigned a non-negative real number P(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 dependencies between them are derived as theorems.
Self-test questions
1. What is the name of the numerical characteristic of the possibility of an event occurring?
2. What is the probability of an event?
3. What is the probability of a reliable event?
4. What is the probability of an impossible event?
5. What are the limits of the probability of a random event?
6. What are the limits of the probability of any event?
7. What definition of probability is called classical?
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 performed 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 “digital” 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 a complete group are pairwise incompatible, then the test will result in 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."
Each of the possible results of the test (the test consists of removing a ball from an urn) will be called 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 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 called 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;
P- 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 that elementary events are pairwise incompatible. The set of all elementary events that can occur in a test is called space of elementary eventsΩ (Greek capital 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.
Probability theory is a mathematical science that studies the patterns of random events. A probabilistic experiment (test, observation) is an experiment whose result cannot be predicted in advance. In this experiment, any result (outcome) is event.
The event may be reliable(always occurs as a result of a test); impossible(obviously does not occur during testing); random(may or may not happen under the conditions of this experiment).
An event that cannot be broken down into simpler events is called elementary. An event presented as a combination of several elementary events is called complex(the company did not suffer losses - profit can be positive or equal to zero).
Two events that cannot occur simultaneously (increase in taxes - increase in disposable income; increase in investment - decrease in risk) are called incompatible.
In other words, two events are incompatible if the occurrence of one of them excludes the occurrence of the other. Otherwise they are joint(increase in sales volume - increase in profits). The events are called opposite, if one of them occurs if and only if the other does not occur (the product is sold - the product is not sold).
Probability of event – This is a numerical measure that is introduced to compare events according to the degree of possibility of their occurrence.
Classic definition of probability. Probability R(A) events A is called the number ratio m equally possible elementary events (outcomes) favorable to the occurrence of the event A, to the total number n all possible elementary outcomes of this experiment:
The following basic properties of probability follow from the above:
1.0 £ R(A) £ 1.
2. Probability of a certain event A equals 1: R(A) = 1.
3. The probability of an impossible event A is 0: R(A) = 0.
4. If events A And IN are incompatible, then R(A + IN) = R(A) + R(IN); if events A And IN are joint, then R(A + IN) = R(A) + R(IN) - R(A . B).(R(A . B) is the probability of the joint occurrence of these events).
5. If A and opposite events, then R() = 1 - R(A).
If the probability of one event occurring does not change the probability of another occurring, then such events are called independent.
When directly calculating the probabilities of events characterized by a large number of outcomes, one should use combinatorics formulas. To study a group of events (hypotheses)
the formulas of total probability, Bayes and Bernoulli are applied ( n independent tests - repetition of experiments).
At statistical determination of probability events A under n refers to the total number of tests actually performed in which the event A met exactly m once. In this case the relation m/n called relative frequency (frequency) W n(A) occurrence of the event A V n tests performed.
When determining the probability by method of expert assessments under n refers to the number of experts (specialists in a given field) interviewed regarding the possibility of an event occurring A. Wherein m of which they claim that the event A will happen.
The concept of a random event is not enough to describe the results of observations of quantities that have a numerical expression. For example, when analyzing the financial result of an enterprise, they are primarily interested in its size. Therefore, the concept of a random event is complemented by the concept of a random variable.
Under random variable(SV) is understood as a quantity that, as a result of observation (testing), takes on one of a possible set of its values, unknown in advance and depending on random circumstances. For each elementary event, SV has a single meaning.
There are discrete and continuous SVs. For discrete SV the set of its possible values is finite or countable, i.e. SV takes on individual isolated values that can be listed in advance, with certain probabilities. For continuous SV, the set of its possible values is infinite and uncountable, for example, all numbers of a given interval, i.e. possible values of SV cannot be listed in advance and continuously fill a certain gap.
Examples of random variables: X- daily number of customers in the supermarket (discrete SV); Y- the number of children born during the day in a certain administrative center (discrete SV); Z- coordinate of the point of impact of an artillery shell (continuous NE).
Many SVs considered in economics have such a large number of possible values that it is more convenient to represent them in the form of continuous SVs. For example, exchange rates, household income, etc.
To describe the SV, it is necessary to establish a relationship between all possible values of the SV and their probabilities. This ratio will be called law of distribution of SV. For a discrete SV it can be specified tabularly, analytically (in the form of a formula) or graphically. For example, tabular for SV X