The probability of an event is equal to one. Classical and statistical definition of probability. Basic theoretical information
Different definitions of the probability of a random event
Probability theory- mathematical science, which, according to the probabilities of some events, allows us to evaluate the probabilities of other events associated with the first.
Confirmation that the concept of "probability of an event" has no definition is the fact that in the theory of probability there are several approaches to explaining this concept:
Classical definition of probability random event .
The probability of an event is equal to the ratio of the number of outcomes of the experience favorable to the event to the total number of outcomes of the experiment.
Where
The number of favorable outcomes of the experience;
Total number of experiences.
The outcome of the experience is called favorable for an event, if an event appeared during this outcome of the experience. For example, if the event is the appearance of a card of red suit, then the appearance of an ace of diamonds is an outcome favorable to the event.
Examples.
1) The probability of getting 5 points on the edge of the cube is equal, since the cube can fall any of the 6 edges up, and 5 points are on only one edge.
2) The probability of the coat of arms falling out with a single toss of the coin - since the coin can fall with the coat of arms or tails - two outcomes of the experience, and the coat of arms is depicted only on one side of the coin.
3) If there are 12 balls in the urn, of which 5 are black, then the probability of removing the black ball is, since there are 12 mushroom outcomes in total, and 5 favorable ones
Comment. The classical definition of probability is applicable under two conditions:
1) all outcomes of the experiment must be equally probable;
2) the experience must have a finite number of outcomes.
In practice, it is difficult to prove that events are equally probable: for example, when performing an experiment with tossing a coin, the result of the experiment can be influenced by such factors as the asymmetry of the coin, the effect of its shape on the aerodynamic characteristics of the flight, atmospheric conditions, etc., in addition, there are experiments with an infinite number of outcomes.
Example ... The child throws the ball, and the maximum distance he can throw the ball is 15 meters. Find the probability that the ball will fly past the 3 m mark.
Solution.The desired probability is proposed to be considered as the ratio of the length of the segment located beyond the 3 m mark (favorable area) to the length of the entire segment (all possible outcomes):
Example. A point is randomly thrown into a circle of radius 1. What is the probability that a point will fall into a square inscribed in a circle?
Solution.The probability that a point will fall into a square is understood in this case as the ratio of the area of the square (favorable area) to the area of the circle (the total area of the figure where the point is thrown):
The diagonal of the square is 2 and is expressed in terms of its side according to the Pythagorean theorem:
Similar reasoning is carried out in space: if a point is randomly selected in the body of volume, then the probability that the point will be in a part of the body of volume is calculated as the ratio of the volume of the favorable part to the total volume of the body:
Combining all cases, we can formulate a rule for calculating the geometric probability:
If a point is randomly selected in some area, then the probability that the point will be in a part of this area is equal to:
, where
Indicates the measure of the area: in the case of a segment, this is the length, in the case of a flat area, this is the area, in the case of a spatial body, this is the volume, on the surface - the surface area, on the curve - the length of the curve.
An interesting application of the concept of geometric probability is the encounter problem.
Task. (About meeting)
Two students made an appointment, for example, at 10 o'clock in the morning on the following conditions: each one comes at any time during an hour from 10 to 11 and waits for 10 minutes, after which he leaves. What is the likelihood of a meeting?
Solution.Let us illustrate the conditions of the problem as follows: on the axis we plot the time that passes for the first of those encountered, and on the axis, the time that passes for the second. Since the experiment lasts one hour, we will postpone segments of length 1 along both axes. Moments of time when those encountered came at the same time are interpreted by the diagonal of the square.
Let the first come at some point in time. Students will meet if the arrival time of the second at the meeting point is between
Arguing in this way for any moment in time, we get that the time area interpreting the possibility of meeting ("intersection of times" of being in the right place of the first and second students) is between two straight lines: and ... The probability of a meeting is determined by the geometric probability formula:
In 1933 Kolmogorov A.M. (1903 - 1987) proposed an axiomatic approach to the construction and presentation of the theory of probability, which has become generally accepted at the present time. When constructing a theory of probability as a formal axiomatic theory, it is required not only to introduce a basic concept - the probability of a random event, but also to describe its properties using axioms (statements that are intuitively correct, accepted without proof).
Such statements are statements similar to the properties of the relative frequency of occurrence of an event.
The relative frequency of occurrence of a random event is the ratio of the number of occurrences of an event in tests to the total number of tests performed:
Obviously, for a reliable event, for an impossible event, for inconsistent events, the following is true:
Example. Let us illustrate the last statement. Have cards taken out of a deck of 36 cards. Let the event mean the appearance of diamonds, the event means the appearance of hearts, and the event means the appearance of a card of red suit. Obviously, events are inconsistent. When a red suit appears, we put a mark near the event, when diamonds appear - near the event, and when worms appear - near the event. Obviously, a mark near an event will be placed if and only if a mark is placed near an event or near an event, i.e. ...
Let's call the probability of a random event the number associated with the event according to the following rule:
For inconsistent events and
So,
|
Relative frequency |
1. Statement of the main theorems and formulas of probabilities: addition theorem, conditional probability, multiplication theorem, independence of events, formula of total probability.
Goals: creation of favorable conditions for the introduction of the concept of the probability of an event; acquaintance with the main theorems and formulas of probability theory; introduce the formula for the total probability.
Course of the lesson:
Random experiment (experience) is called a process in which different outcomes are possible, and it is impossible to predict in advance what the result will be. Possible mutually exclusive outcomes of the experience are called its elementary events ... We denote the set of elementary events by W.
By a random event is called an event about which it is impossible to say in advance whether it will happen as a result of experience or not. Each random event A, which occurred as a result of the experiment, can be associated with a group of elementary events from W. Elementary events that make up this group are called favorable for the occurrence of the event A.
The set W can also be viewed as a random event. Since it includes all elementary events, it will definitely happen as a result of experience. Such an event is called reliable .
If there are no favorable elementary events from W for a given event, then it cannot occur as a result of the experiment. Such an event is called impossible.
Events are called equally possible if the test results in equal opportunities for these events to occur. Two random events are called opposite , if, as a result of the experiment, one of them occurs if and only if the other does not happen. The event opposite to event A is designated.
Events A and B are called inconsistent if the appearance of one of them excludes the appearance of the other. Events А 1, А 2, ..., А n are called pairwise incompatible, if any two of them are inconsistent. Events А 1, А 2, ..., Аn form complete system pairwise incompatible events if, as a result of the test, one and only one of them will necessarily occur.
The sum (combination) of eventsА 1, А 2, ..., А n is called an event С, which consists in the fact that at least one of the events А 1, А 2, ..., А n has occurred.The sum of events is denoted as follows:
C = A 1 + A 2 + ... + A n.
By the product (intersection) of eventsА 1, А 2, ..., А n is called such an event P, which consists in the fact that all events А 1, А 2, ..., А n occurred simultaneously. The product of events is indicated by
The probability P (A) in the theory of probability acts as a numerical characteristic of the degree of possibility of the occurrence of a certain random event A with multiple repetitions of tests.
Let's say that with 1000 rolls of the dice, the number 4 falls out 160 times. The ratio 160/1000 = 0.16 shows the relative frequency of occurrence of the digit 4 in this series of tests. More generally the frequency of the random event And when conducting a series of experiments, the ratio of the number of experiments in which this event occurred to the total number of experiments is called:
where Р * (А) - frequency of event А; m is the number of experiments in which event A occurred; n is the total number of experiments.
The probability of a random event And they call a constant number, around which the frequencies of a given event are grouped as the number of experiments increases ( statistical determination of the probability of an event ). The probability of a random event is denoted by P (A).
Naturally, no one will ever be able to carry out an unlimited number of tests in order to determine the probability. This is not even necessary. In practice, the probability can be taken as the frequency of an event with a large number of trials. So, for example, from the statistical patterns of birth established over many years of observation, the probability of the event that the newborn will be a boy is estimated at 0.515.
If during the test there are no reasons due to which one random event would appear more often than others ( equally possible events), the probability can be determined based on theoretical considerations. For example, let us find out, in the case of a coin toss, the frequency of the coat of arms falling (event A). it was shown by various experimenters in several thousand tests that the relative frequency of such an event takes on values close to 0.5. Considering that the appearance of the coat of arms and the opposite side of the coin (event B) are equally possible events, if the coin is symmetrical, the judgment P (A) = P (B) = 0.5 could be made without determining the frequency of these events. A different definition of probability is formulated on the basis of the concept of “equal opportunity” of events.
Let the considered event A occur in m cases, which are called favorable to A, and does not occur in the remaining n-m, unfavorable to A.
Then the probability of event A is equal to the ratio of the number of favorable elementary events to their total number(classical definition of the probability of an event):
where m is the number of elementary events favorable to event A; n - The total number of elementary events.
Let's look at a few examples:
Example # 1:The urn contains 40 balls: 10 black and 30 white. Find the probability that the ball chosen at random will be black.
The number of favorable cases is equal to the number of black balls in the urn: m = 10. The total number of equally possible events (taking out one ball) is equal to the total number of balls in the urn: n = 40. These events are inconsistent, since one and only one ball is taken out. P (A) = 10/40 = 0.25
Example # 2:find the probability of getting an even number when throwing a dice.
When throwing a dice, six equally possible inconsistent events are realized: the appearance of one number: 1,2,3,4,5 or 6, i.e. n = 6. favorable cases are the occurrence of one of the digits 2,4 or 6: m = 3. the desired probability P (A) = m / N = 3/6 = ½.
As we can see from the definition of the probability of an event, for all events
0 < Р(А) < 1.
Obviously, the probability of a certain event is 1, the probability of an impossible event is 0.
The theorem of addition of probabilities: the probability of occurrence of one (no matter what) event from several incompatible events is equal to the sum of their probabilities.
For two inconsistent events A and B, the probabilities of these events are equal to the sum of their probabilities:
P (A or B) = P (A) + P (B).
Example # 3:find the probability of falling out 1 or 6 when throwing a dice.
Event A (roll 1) and B (roll 6) are equally possible: P (A) = P (B) = 1/6, therefore P (A or B) = 1/6 + 1/6 = 1/3
The addition of probabilities is valid not only for two, but also for any number of inconsistent events.
Example # 4:the urn contains 50 balls: 10 white, 20 black, 5 red and 15 blue. Find the probability of a white, or black, or red ball appearing in a single operation of removing the ball from the urn.
The probability of taking out the white ball (event A) is P (A) = 10/50 = 1/5, the black ball (event B) is P (B) = 20/50 = 2/5 and the red ball (event C) is P (C) = 5/50 = 1/10. From here, using the formula for adding the probabilities, we get P (A or B or C) = P (A) + P (B) = P (C) = 1/5 + 2/5 + 1/10 = 7/10
The sum of the probabilities of two opposite events, as follows from the theorem of addition of probabilities, is equal to one:
P (A) + P () = 1
In the above example, taking out a white, black and red ball will be the event А 1, Р (А 1) = 7/10. Opposite event 1 is the reaching of the blue ball. Since there are 15 blue balls, and the total number of balls is 50, we get P (1) = 15/50 = 3/10 and P (A) + P () = 7/10 +3/10 = 1.
If events А 1, А 2, ..., А n form a complete system of pairwise incompatible events, then the sum of their probabilities is equal to 1.
In the general case, the probability of the sum of two events A and B is calculated as
P (A + B) = P (A) + P (B) - P (AB).
Probability multiplication theorem:
Events A and B are called independent , if the probability of occurrence of event A does not depend on whether event B has occurred or not, and vice versa, the probability of occurrence of event B does not depend on whether event A has occurred or not.
The probability of joint occurrence of independent events is equal to the product of their probabilities... For two events P (A and B) = P (A) P (B).
Example: In one urn there are 5 black and 10 white balls, in the other 3 black and 17 white balls. Find the probability that when the balls are first removed from each urn, both balls will turn out to be black.
Solution: the probability of pulling out a black ball from the first urn (event A) - P (A) = 5/15 = 1/3, a black ball from the second urn (event B) - P (B) = 3/20
P (A and B) = P (A) P (B) = (1/3) (3/20) = 3/60 = 1/20.
In practice, the probability of event B often depends on whether some other event A has occurred or not. In this case, they talk about conditional probability , i.e. the probability of event B, provided that event A has occurred. The conditional probability is denoted by P (B / A).
ChapterI... RANDOM EVENTS. PROBABILITY
1.1. Regularity and randomness, random variability in the exact sciences, in biology and medicine
Probability theory is a branch of mathematics that studies patterns in random phenomena. A random phenomenon is a phenomenon that, with repeated reproduction of the same experience, can proceed somewhat differently each time.
Obviously, there is not a single phenomenon in nature in which elements of chance are not present to one degree or another, but in different situations we take them into account in different ways. So, in a number practical tasks they can be neglected and considered instead of a real phenomenon its simplified scheme - "model", assuming that under the given conditions of experience the phenomenon proceeds in a quite definite way. At the same time, the most important, decisive factors that characterize the phenomenon are highlighted. It is this scheme of studying phenomena that is most often used in physics, technology, mechanics; this is how the basic pattern comes to light , characteristic of this phenomenon and making it possible to predict the result of the experiment according to the given initial conditions. And the influence of random, secondary, factors on the result of the experiment is taken into account here by random measurement errors (we will consider the method of calculating them below).
However, the described classical scheme of the so-called exact sciences is ill-suited for solving many problems in which numerous, closely intertwined random factors play a noticeable (often decisive) role. Here the random nature of the phenomenon comes to the fore, which can no longer be neglected. This phenomenon must be studied precisely from the point of view of the laws inherent in it as a random phenomenon. In physics, examples of such phenomena are Brownian motion, radioactive decay, a number of quantum mechanical processes, etc.
The subject of study by biologists and physicians is a living organism, the origin, development and existence of which is determined by very many and varied, often random external and internal factors. That is why the phenomena and events of the living world are also in many ways accidental in nature.
Elements of uncertainty, complexity, multi-causality inherent in random phenomena necessitate the creation of special mathematical methods for studying these phenomena. The development of such methods, the establishment of specific patterns inherent in random phenomena, are the main tasks of the theory of probability. It is characteristic that these patterns are fulfilled only when random phenomena are massive. And individual characteristics individual cases seem to be mutually canceled out, and the average result for a mass of random phenomena is no longer random, but quite natural . To a large extent, this circumstance was the reason for the widespread use of probabilistic research methods in biology and medicine.
Let's consider the basic concepts of the theory of probability.
1.2. The probability of a random event
Each science that develops a general theory of a certain range of phenomena is based on a number of basic concepts. For example, in geometry, these are the concepts of a point, a straight line; in mechanics - the concepts of force, mass, velocity, etc. Basic concepts exist in the theory of probability, one of them is a random event.
A random event is any phenomenon (fact) that, as a result of experience (testing), may or may not occur.
Random events are indicated by letters A, B, C... and so on. Here are some examples random events:
A–The falling of the eagle (coat of arms) when a standard coin is tossed;
V- the birth of a girl in this family;
WITH- the birth of a child with a predetermined body weight;
D- the emergence of an epidemic disease in a given region in a certain period of time, etc.
The main quantitative characteristic of a random event is its probability. Let A- some random event. The probability of a random event A is a mathematical quantity that determines the likelihood of its occurrence. It is denoted R(A).
Consider two main methods for determining this value.
The classical definition of the probability of a random event usually based on the results of the analysis of speculative experiments (tests), the essence of which is determined by the condition of the task. In this case, the probability of a random event P (A) is equal to:
where m- the number of cases favorable for the occurrence of the event A; n- the total number of equally possible cases.
Example 1. A laboratory rat is placed in a maze, in which only one of the four possible ways leads to food rewards. Determine the likelihood of the rat taking this path.
Solution: by the condition of the problem, out of four equally possible cases ( n= 4) event A(the rat finds food)
only one favors, i.e. m= 1 Then R(A) = R(the rat finds food) = = 0.25 = 25%.
Example 2. There are 20 black and 80 white balls in the urn. One ball is taken out of it at random. Determine the probability that this ball will be black.
Solution: the number of all balls in the urn is the total number of equally possible cases n, i.e. n = 20 + 80 = 100, of which event A(removing the black ball) is possible only at 20, i.e. m= 20. Then R(A) = R(h. w.) = = 0.2 = 20%.
Let us list the properties of probability following from its classical definition - formula (1):
1. The probability of a random event is a dimensionless quantity.
2. The probability of a random event is always positive and less than one, that is, 0< P (A) < 1.
3. The probability of a reliable event, that is, an event that will necessarily occur as a result of the experiment ( m = n) is equal to one.
4. The probability of an impossible event ( m= 0) is equal to zero.
5. The probability of any event is not a negative value and does not exceed one:
0 £ P (A) £ 1.
Statistical determination of the probability of a random event is used when it is impossible to use the classical definition (1). This is often the case in biology and medicine. In this case, the probability R(A) is determined by summarizing the results of actually carried out series of tests (experiments).
Let us introduce the concept of the relative frequency of occurrence of a random event. Let there be a series consisting of N experiments (number N can be selected in advance); event of interest to us A happened in M of them ( M < N). The ratio of the number of experiments M in which this event occurred, to the total number of experiments carried out N called the relative frequency of occurrence of a random event A in this series of experiments - R* (A)
R*(A) = .
It has been experimentally established that if a series of tests (experiments) are carried out under the same conditions and in each of them the number N is large enough, then the relative frequency exhibits the property of stability : from series to series it changes little , approaching with an increase in the number of experiments to a certain constant value . It is taken for the statistical probability of a random event A:
R(A)= lim, for N , (2)
So, the statistical probability R(A) random event A is called the limit to which the relative frequency of occurrence of this event tends with an unlimited increase in the number of tests (with N → ∞).
The approximate statistical probability of a random event is equal to the relative frequency of occurrence of this event for a large number of tests:
R(A)≈ P *(A)= (for large N) (3)
For example, in experiments on tossing a coin, the relative frequency of the emblem falling out with 12,000 tosses turned out to be 0.5016, and with 24,000 tosses - 0.5005. According to formula (1):
P(coat of arms) = = 0.5 = 50%
Example . During a medical examination of 500 people, 5 of them were found to have a tumor in the lungs (o. l.). Determine the relative frequency and likelihood of this disease.
Solution: by the condition of the problem M = 5, N= 500, relative frequency R* (o. l.) = M/N= 5/500 = 0.01; insofar as N is large enough, it can be assumed with good accuracy that the probability of the presence of a tumor in the lungs is equal to the relative frequency of this event:
R(o. l.) = R* (o. l.) = 0.01 = 1%.
The previously enumerated properties of the probability of a random event are retained during the statistical determination of this quantity.
1.3. Types of random events. Fundamental Theorems of Probability Theory
All random events can be divided into:
¾ incompatible;
¾ independent;
¾ dependent.
Each type of event has its own characteristics and theorems of probability theory.
1.3.1. Incompatible random events. Probability addition theorem
Random events (A, B, C,D...) are called inconsistent , if the occurrence of one of them excludes the occurrence of other events in the same trial.
Example 1 . A coin tossed. When it falls, the appearance of the "coat of arms" excludes the appearance of "tails" (an inscription that determines the price of the coin). The events “the coat of arms fell out” and “the tails fell out” were inconsistent.
Example 2 . A student receiving a grade of “2” or “3” or “4” or “5” on one exam are inconsistent events, since one of these grades excludes the other on the same exam.
For inconsistent random events, addition theorem: probability of occurrence one, but all the same which, of several incompatible events A1, A2, A3 ... Ak is equal to the sum of their probabilities:
P (A1 or A2 ... or Ak) = P (A1) + P (A2) + ... + P (Ak). (4)
Example 3. The urn contains 50 balls: 20 white, 20 black and 10 red. Find the probability of a white (event A) or a red ball (event V) when the ball is taken out of the urn at random.
Solution: P(A or B)= P(A)+ P(V);
R(A) = 20/50 = 0,4;
R(V) = 10/50 = 0,2;
R(A or V)= P(b. w. or k. w.) = 0,4 + 0,2 = 0,6 = 60%.
Example 4 . There are 40 children in the class. Of these, at the age from 7 to 7.5 years, 8 boys ( A) and 10 girls ( V). Find the likelihood of having children of this age in the classroom.
Solution: P(A)= 8/40 = 0.2; R(V) = 10/40 = 0,25.
P (A or B) = 0.2 + 0.25 = 0.45 = 45%
The next important concept is full group of events: several incompatible events form a complete group of events if as a result of each test, only one of the events of this group and no other can appear.
Example 5 . The shooter fired a shot at the target. One of the following events will surely happen: hitting the top ten, the nine, the eight, .., the one, or a miss. These 11 inconsistent events form a complete group.
Example 6 . At the university exam, a student can receive one of the following four grades: 2, 3, 4 or 5. These four inconsistent events also form a complete group.
If inconsistent events A1, A2 ... Ak form a complete group, then the sum of the probabilities of these events is always equal to one:
R(A1)+ P(A2)+ ... P(Ak) = 1, (5)
This statement is often used in solving many applied problems.
If two events are the only possible and incompatible, then they are called opposite and denote A and . Such events make up a complete group, so the sum of their probabilities is always equal to one:
R(A)+ P() = 1. (6)
Example 7. Let R(A) - the likelihood of death in a certain disease; it is known and equal to 2%. Then the probability of a successful outcome in this disease is 98% ( R() = 1 – R(A) = 0.98), since R(A) + R() = 1.
1.3.2. Independent random events. Probability multiplication theorem
Random events are called independent if the occurrence of one of them does not in any way affect the likelihood of other events occurring.
Example 1 . If there are two or more urns with colored balls, then removing any ball from one urn will in no way affect the likelihood of removing other balls from the remaining urns.
For independent events, it is true probability multiplication theorem: the probability of a joint(simultaneous)the occurrence of several independent random events is equal to the product of their probabilities:
P (A1 and A2 and A3 ... and Ak) = Р (А1) ∙ Р (А2) ∙ ... ∙ Р (Аk). (7)
Joint (simultaneous) occurrence of events means that events occur and A1, and A2, and A3… and Ak .
Example 2 . There are two urns. One contains 2 black and 8 white balls, the other contains 6 black and 4 white balls. Let the event A- choosing at random a white ball from the first urn, V- from the second. What is the probability of choosing at random from these urns at the same time using a white ball, that is, what is R (A and V)?
Solution: the probability of getting the white ball from the first urn
R(A) = = 0.8 from the second - R(V) = = 0.4. The probability of simultaneously getting a white ball from both urns is
R(A and V) = R(A)· R(V) = 0,8∙ 0,4 = 0,32 = 32%.
Example 3 A diet with low iodine content causes an enlargement of the thyroid gland in 60% of animals in a large population. The experiment requires 4 enlarged glands. Find the probability that 4 randomly selected animals will have an enlarged thyroid gland.
Solution: Random event A- choosing at random an animal with an enlarged thyroid gland. According to the condition of the problem, the probability of this event is R(A) = 0.6 = 60%. Then the probability of the joint occurrence of four independent events - choosing at random 4 animals with an enlarged thyroid gland - will be equal to:
R(A 1 and A 2 and A 3 and A 4) = 0,6 ∙ 0,6 ∙0,6 ∙ 0,6=(0,6)4 ≈ 0,13 = 13%.
1.3.3. Dependent events. Probability multiplication theorem for dependent events
Random events A and B are called dependent if the appearance of one of them, for example, A changes the probability of the appearance of another event - B. Therefore, for dependent events, two probability values are used: unconditional and conditional probabilities .
If A and V dependent events, then the probability of the occurrence of the event V first (i.e. before the event A) is called unconditional probability this event and is designated R(V). The probability of an event occurring V provided that the event A has already happened, called conditional probability events V and denoted R(V/A) or RA(V).
Unconditional have a similar meaning - R(A) and conditional - R(A / B) the probabilities for the event A.
The theorem for multiplying the probabilities for two dependent events: the probability of the simultaneous occurrence of two dependent events A and B is equal to the product of the unconditional probability of the first event by the conditional probability of the second:
R(A and B)= P(A)∙ R(B / A) , (8)
A, or
R(A and B)= P(V)∙ R(A / B), (9)
if the event comes first V.
Example 1. There are 3 black balls and 7 white balls in the urn. Find the probability that from this urn one after another (and the first ball is not returned to the urn) 2 white balls will be taken out.
Solution: the probability of getting the first white ball (event A) is 7/10. After it is taken out, 9 balls remain in the urn, 6 of them are white. Then the probability of the appearance of the second white ball (event V) is equal to R(V/A) = 6/9, and the probability of getting two white balls in a row is
R(A and V) = R(A)∙R(V/A) = = 0,47 = 47%.
The above theorem of multiplication of probabilities for dependent events can be generalized to any number of events. In particular, for three events related to each other:
R(A and V and WITH)= P(A)∙ R(B / A)∙ R(C / AB). (10)
Example 2. In two kindergartens, each of which attends 100 children, an outbreak occurred infectious disease... The proportions of cases are, respectively, 1/5 and 1/4, with 70% in the first institution, and in the second - 60% of cases - children under 3 years of age. One child is selected at random. Determine the likelihood that:
1) the selected child belongs to the first kindergarten (event A) and sick (event V).
2) the child is selected from the second kindergarten(event WITH), sick (event D) and over 3 years old (event E).
Solution. 1) the desired probability -
R(A and V) = R(A) ∙ R(V/A) = = 0,1 = 10%.
2) the required probability:
R(WITH and D and E) = R(WITH) ∙ R(D/C) ∙ R(E/CD) = 
= 5%.
1.4. Bayes formula
If the probability of joint occurrence of dependent events A and V does not depend on the order in which they occur, then R(A and V)= P(A)∙ R(B / A)= P(V) × R(A / B). In this case, the conditional probability of one of the events can be found knowing the probabilities of both events and the conditional probability of the second:
R(B / A) = (11)
The Bayesian formula is a generalization of this formula to the case of many events.
Let " n»Inconsistent random events Н1, Н2, ..., Нn, form a complete group of events. The probabilities of these events are - R(H1), R(H2), …, R(Nn) are known and since they form a complete group, then = 1.
Some random event A related to events Н1, Н2, ..., Нn, and the conditional probabilities of the occurrence of the event are known A with each of the events Ni, i.e., known R(A / H1), R(A / H2), …, R(A / Hn). In this case, the sum of conditional probabilities R(A / Hi) may not be equal to one, i.e. 
≠ 1.
Then the conditional probability of the occurrence of the event Ni when the event is implemented A(i.e., provided that the event A happened) is determined by the Bayes formula :
Moreover, for these conditional probabilities 
.
Bayes' formula has found wide application not only in mathematics, but also in medicine. For example, it is used to calculate the probabilities of certain diseases. So if N 1,…, Nn- the estimated diagnoses for a given patient, A- some sign related to them (a symptom, a certain indicator of a blood test, urine analysis, a detail of an X-ray, etc.), and conditional probabilities R(A / Hi) manifestations of this symptom with each diagnosis Ni (i = 1,2,3,…n) are known in advance, then Bayes' formula (12) allows calculating the conditional probabilities of diseases (diagnoses) R(Ni/A) after it has been established that the characteristic feature A is present in the patient.
Example 1. At the initial examination of the patient, 3 diagnoses are assumed N 1, N 2, N 3. Their probabilities, according to the doctor, are distributed as follows: R(N 1) = 0,5; R(N 2) = 0,17; R(N 3) = 0.33. Therefore, the first diagnosis seems to be the most probable in advance. To clarify it, for example, a blood test is prescribed, in which an increase in ESR is expected (event A). It is known in advance (based on research results) that the probabilities of an increase in ESR in case of suspected diseases are equal to:
R(A/N 1) = 0,1; R(A/N 2) = 0,2; R(A/N 3) = 0,9.
The resulting analysis recorded an increase in ESR (event A happened). Then the calculation according to the Bayes formula (12) gives the values of the probabilities of the alleged diseases with an increased ESR value: R(N 1/A) = 0,13; R(N 2/A) = 0,09;
R(N 3/A) = 0.78. These figures show that, taking into account laboratory data, the most realistic is not the first, but the third diagnosis, the probability of which has now turned out to be quite high.
The given example is the simplest illustration of how the Bayesian formula can be used to formalize the logic of a doctor when making a diagnosis and, thanks to this, create methods of computer diagnostics.
Example 2. Determine the probability that estimates the risk of perinatal * mortality of a child in women with anatomically narrow pelvis.
Solution: let the event N 1 - successful childbirth. According to clinical reports, R(N 1) = 0.975 = 97.5%, then if H2- the fact of perinatal mortality, then R(N 2) = 1 – 0,975 = 0,025 = 2,5 %.
We denote A- the fact that a woman in labor has a narrow pelvis. From the studies carried out, the following are known: a) R(A/N 1) - the likelihood of a narrow pelvis with favorable childbirth, R(A/N 1) = 0.029, b) R(A/N 2) - the likelihood of a narrow pelvis with perinatal mortality,
R(A/N 2) = 0.051. Then the desired probability of perinatal mortality with a narrow pelvis in a woman in labor is calculated using the Bays formula (12) and is equal to:
Thus, the risk of perinatal mortality in anatomically narrow pelvis is significantly higher (almost twice) than the average risk (4.4% versus 2.5%).
Such calculations, usually performed with the help of a computer, underlie the methods of forming groups of patients at increased risk associated with the presence of one or another aggravating factor.
Bayes' formula is very useful for evaluating many other biomedical situations, which will become obvious when solving the problems given in the manual.
1.5. On random events with probabilities close to 0 or 1
When solving many practical problems, one has to deal with events, the probability of which is very small, i.e., close to zero. Based on experience with such events, the following principle has been adopted. If a random event has a very low probability, then in practice it can be assumed that in a single test it will not occur, in other words, the possibility of its occurrence can be neglected. The answer to the question of how small this probability should be is determined by the essence of the problems being solved, by how important the prediction result is for us. For example, if the probability that the parachute will not open during the jump is 0.01, then the use of such parachutes is unacceptable. However, the likelihood of a long-distance train arriving late, equal to the same 0.01, makes us almost sure that it will arrive on time.
A sufficiently small probability at which (in a given specific problem) an event can be considered practically impossible is called level of significance. In practice, the significance level is usually taken equal to 0.01 (one percent significance level) or 0.05 (five percent significance level), much less often it is taken equal to 0.001.
The introduction of the significance level allows us to assert that if some event A almost impossible, then the opposite event - practically reliable, that is, for him R() » 1.
ChapterII... RANDOM VALUES
2.1. Random variables, their types
In mathematics, a quantity is a general name for various quantitative characteristics of objects and phenomena. Length, area, temperature, pressure, etc. are examples of different quantities.
The value that takes different numerical values under the influence of random circumstances, called a random variable... Examples of random variables: number of patients at a doctor's appointment; exact dimensions internal organs people, etc.
Distinguish between discrete and continuous random variables .
A random variable is called discrete if it only takes certain values, separated from each other, which can be set and enumerated.
Examples of discrete random variables are:
- the number of students in the classroom - can only be a positive integer number: 0,1,2,3,4… .. 20… ..;
- the number that appears on the upper edge when the dice is thrown - can only take whole values from 1 to 6;
- relative frequency of hitting the target with 10 shots - its values: 0; 0.1; 0.2; 0.3 ... 1
- the number of events that occur over the same time intervals: heart rate, number of ambulance calls per hour, number of operations per month with fatal outcomes, etc.
A random variable is called continuous if it can take on any values within a certain interval, which sometimes has sharply defined boundaries, and sometimes not.*. Continuous random variables include, for example, body weight and height of adults, body weight and brain volume, the quantitative content of enzymes in healthy people, the size of blood cells, R H blood, etc.
Concept random variable plays a decisive role in modern probability theory, which has developed special techniques for the transition from random events to random variables.
If a random variable depends on time, then we can talk about a random process.
2.2. Distribution law of a discrete random variable
To give a complete description of a discrete random variable, it is necessary to indicate all its possible values and their probabilities.
The correspondence between the possible values of a discrete random variable and their probabilities is called the distribution law of this quantity.
We denote the possible values of the random variable X across Xi, and the corresponding probabilities - through Ri *. Then the distribution law of a discrete random variable can be specified in three ways: in the form of a table, graph or formula.
In a table called a distribution series, lists all possible values of a discrete random variable X and the corresponding probabilities R(X):
X
…..
…..
P(X)
…..
…..
Moreover, the sum of all probabilities Ri must be equal to one (normalization condition):
Ri = p1 + p2 + ... + pn = 1. (13)
Graphically the law is represented by a broken line, which is usually called the distribution polygon (Fig. 1). Here, all possible values of the random variable are plotted along the horizontal axis Xi,
,
and along the vertical axis - the corresponding probabilities Ri
Analytically the law is expressed by the formula. For example, if the probability of hitting a target in one shot is R, then the probability of hitting the target 1 time at n shots is given by the formula R(n) = n qn-1 × p, where q= 1 - p- the probability of a miss with one shot.
2.3. The distribution law of a continuous random variable. Probability distribution density
For continuous random variables, it is impossible to apply the distribution law in the forms given above, since such a quantity has an infinite ("uncountable") set of possible values that completely fill a certain interval. Therefore, it is impossible to compile a table in which all its possible values are listed, or to build a distribution polygon. In addition, the probability of any specific value is very small (close to 0) *. At the same time, different areas (intervals) of possible values of a continuous random variable are not equally probable. Thus, in this case, a certain distribution law is in effect, although not in the same sense.
Consider a continuous random variable X, the possible values of which completely fill a certain interval (a, b)**. The probability distribution law for such a value should allow one to find the probability of its value falling into any set interval (x1, x2) lying inside ( a,b), Fig. 2.
This probability is denoted R(x1< Х < х2
), or
R(x1£ X£ x2).
Consider first a very small range of values X- from X before ( x +DX); see fig. 2. Low probability dR the fact that the random variable X will take some value from the interval ( x, x +DX), will be proportional to the value of this interval DX:dR~ DX, or, introducing the proportionality coefficient f which itself may depend on X, we get:
dP =f(X) × D x =f(x) × dx (14)
The function introduced here f(X) is called density of probability distribution random variable X, or, in short, probability density, distribution density... Equation (13) is a differential equation, the solution of which gives the probability of hitting the quantity X in the interval ( x1,x2):
R(x1<X<x2) = f(X) dX. (15)
Graphically the probability R(x1<X<x2) is equal to the area of the curvilinear trapezoid bounded by the abscissa, the curve f(X) and straight lines X = x1 and X = x2(fig. 3). This follows from the geometric meaning of the definite integral (15) Curve f(X) is called the distribution curve.
It follows from (15) that if the function is known f(X), then, by changing the limits of integration, we can find the probability for any intervals of interest to us. Therefore, it is the task of the function f(X) completely determines the distribution law for continuous random variables.
For the probability density f(X) the normalization condition must be satisfied in the form:
f(X) dx = 1, (16)
if it is known that all values X lie in the interval ( a,b), or in the form:
f(X) dx = 1, (17)
if the bounds of the interval for values X definitely vague. The conditions for the normalization of the probability density (16) or (17) are a consequence of the fact that the values of the random variable X lie reliably within ( a,b) or (- ¥, + ¥). From (16) and (17) it follows that the area of the figure bounded by the distribution curve and the abscissa is always equal to 1 .
2.4. Basic numerical characteristics of random variables
The results presented in Sections 2.2 and 2.3 show that a complete characterization of discrete and continuous random variables can be obtained knowing the laws of their distribution. However, in many practically significant situations, they use the so-called numerical characteristics of random variables, the main purpose of these characteristics is to express in a concise form the most significant features of the distribution of random variables. It is important that these parameters represent specific (constant) values that can be estimated using the data obtained in the experiments. Descriptive Statistics deals with these assessments.
A lot of different characteristics are used in probability theory and mathematical statistics, but we will consider only the most used ones. Moreover, only for some of them we will give the formulas by which their values are calculated, in other cases we will leave the calculations to the computer.
Consider position characteristics - mathematical expectation, fashion, median.
They characterize the position of a random variable on the number axis , that is, indicate some approximate value, around which all possible values of the random variable are grouped. Among them, the most important role is played by the mathematical expectation M(X).
Fundamentals of Probability Theory
Plan:
1. Random events
2. The classical definition of probability
3. Calculation of the probabilities of events and combinatorics
4. Geometric probability
Theoretical information
Random events.
Accidental phenomenon- a phenomenon, the outcome of which is not unambiguously determined. 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, the choice of a product in a store is also a random phenomenon, getting an assessment on an exam is a random phenomenon, illness and recovery are random phenomena, etc.
Examples of random phenomena:
~ Shooting is carried out from a gun set at a given angle to the horizon. His hitting the target is accidental, but the hit of the projectile in a certain "fork" is a regularity. You can specify the distance closer and further than which the projectile will not fly. You get some kind of "projectile dispersion fork"
~ The same body is weighed several times. Strictly speaking, each time different results will be obtained, albeit differing by a negligible amount, but different.
~ An airplane flying along the same route has a certain flight corridor, within which the airplane can maneuver, but it will never have strictly the same route
~ An athlete can never run the same distance with the same time. Its results will also fall within a certain numerical range.
Experience, experiment, observation are tests
Trial- observation or fulfillment of a certain set of conditions that are fulfilled repeatedly, moreover, regularly repeated in this and also the sequence, duration, with observance of other identical parameters.
Consider an athlete's execution of a shot at a target. In order for it to be produced, it is necessary to fulfill such conditions as the preparation of the athlete, loading the weapon, aiming, etc. "Hit" and "missed" - events as a result of a shot.
Event- a qualitative test result.
An event may or may not happen. Events are designated in capital Latin letters. For example: D = "Shooter hits the target". S = "White ball removed". K = "A random lottery ticket with no winnings."
Tossing a coin is a challenge. The fall of her "coat of arms" is one event, the fall of her "number" is the second event.
Any test involves the occurrence of several events. Some of them may be necessary for the researcher at a given moment, while others may not be necessary.
The event is called random if under the implementation of a certain set of conditions S it may or may 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 a test result.
~ The shooter shoots at a target divided into four areas. The shot is a test. Hitting a specific target area is an event.
~ There are colored balls in the urn. One ball is taken at random from the urn. Removing the ball from the urn is a test. The appearance of a ball of a certain color is an event.
Types of random events
1. Events are called inconsistent, if the occurrence of one of them excludes the occurrence of other events in the same trial.
~ A part was taken at random from the 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" - inconsistent.
~ Coin 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 inconsistent.
Several events form full group, if at least one of them appears as a result of the test. In other words, the appearance of at least one of the events of the entire group is a reliable event.
In particular, if the events that form a complete group are pairwise inconsistent, then one and only one of these events will appear as a result of the test. This particular case is of greatest interest to us, since it is used below.
~ Two cash lottery tickets purchased. One and only one of the following events is bound to happen:
1. "the prize fell on the first ticket and did not fall on the second",
2. "the prize did not fall on the first ticket and fell on the second",
3. "the winnings fell on both tickets",
4. "no winnings fell on both tickets."
These events form a complete group of pairwise incompatible events,
~ The shooter fired a shot at the target. One of the following two events is bound to 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 none of them is more possible than the other.
~ The appearance of the "coat of arms" and the appearance of an inscription when a coin is tossed 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 the minting does not affect the fallout of one or the other side of the coin.
~ The appearance of a certain number of points on a thrown dice are equally possible events. Indeed, it is assumed that the dice is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of glasses does not affect the fallout of any face.
3. The event is called reliable, if it cannot but happen
4. The event is called not reliable if it can't 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. ...
Classical definition of probability
Probability is one of the basic concepts of probability theory.
There are several definitions of this concept. Here is a definition that is called classical. Next, we point out the weaknesses of this definition and give other definitions that allow us to overcome the shortcomings of the classical definition.
Consider the situation: The box contains 6 identical balls, with 2 red, 3 blue and 1 white. Obviously, the ability to take a colored (i.e., red or blue) ball out of the urn at random is greater than the ability to take out 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 the occurrence of an event.
In the situation under consideration, we denote:
Event A = "Drawing a colored ball".
Each of the possible test results (the test consists in removing the ball from the urn) will be called elementary (possible) outcome and event. Elementary outcomes can be designated by letters with indices at the bottom, 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 inconsistent events (only one ball will necessarily appear) and they are equally possible (the ball is taken out at random, the balls are the same and thoroughly mixed).
Elementary outcomes in which the event of interest to us occurs will be called favorable outcomes this event. In our example, the event is favored A(a colored ball appears) the following 5 outcomes:
Thus, the event A is observed if one, no matter which, of the elementary outcomes that favors A. This is the appearance of any colored ball, of which there are 5 pieces in a box.
In this example, there are 6 elementary outcomes; 5 of them are favorable for the event A. Hence, P (A) = 5/6. This number gives that quantitative estimate of the degree to which a colored ball is likely to appear.
Definition of probability:
The probability of event A is the ratio of the number of favorable outcomes for this event to the total number of all equally possible inconsistent elementary outcomes that form a 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 inconsistent, equally possible, and form a complete group.
The following properties follow from the definition of probability:
1. The probability of a certain 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 the event is impossible, then none of the elementary outcomes of the test favors 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 the following topics, theorems will be presented that allow us to find the probabilities of other events from the known probabilities of some events.
Measured. There are 6 girls and 4 boys in the group of students. What is the probability that the chosen student will be a girl at random? will there be a young man?
p virgins = 6/10 = 0.6 p jun = 4/10 = 0.4
The concept of "probability" in modern, rigorous courses in probability theory is built on a set-theoretic basis. Let's consider some points of this approach.
Let, as a result of the test, one and only one of the events occur: w i(i = 1, 2, .... n). Events w i, - called elementary events (elementary outcomes). O This implies that elementary events are pairwise incompatible. The set of all elementary events that can appear in the test are called space of elementary eventsΩ (capitalized Greek letter omega), and the elementary events themselves are points of this space..
Event A are identified with a subset (of the space Ω), the elements of which are elementary outcomes that favor A; event V is a subset of Ω, the elements of which are outcomes favorable V, etc. Thus, the set of all events that can occur in the test is the set of all subsets of Ω, Ω itself occurs at any outcome of the test, therefore Ω is a reliable event; an empty subset of the space Ω- is an impossible event (it does not occur for any outcome of the test).
Elementary events are selected from among all the events of the topics, "for each of them contains only one element Ω
To every elementary outcome w i match a positive number p i is the probability of this outcome, and the sum of all p i is equal to 1 or with a sign of the sum, this fact will be written in the form of an expression:
By definition, the probability P (A) events A is equal to the sum of the probabilities of elementary outcomes favorable A. Therefore, the probability of a reliable event is equal to one, the impossible is zero, and the arbitrary is enclosed between zero and one.
Consider an important special case, when all outcomes are equally possible, the number of outcomes is equal to 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.
Event probability A is equal to the sum of the probabilities of outcomes favorable A:
P (A) = 1 / n + 1 / n + ... + 1 / n = n · 1 / n = 1
The classical definition of probability is obtained.
There is still axiomatic approach to the concept of "probability". In the system of axioms proposed. Kolmogorov A. N, the 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 determine the probability:
1. Each event A mapped to 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 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 to the dependence between them are deduced as theorems.
For practical activity, it is necessary to be able to compare events according to the degree of their possibility of occurrence. Consider the classic case. The urn contains 10 balls, 8 of them are white, 2 are black. Obviously, the event “a white ball will be removed from the urn” and the event “a black ball will be removed from the urn” have different degrees of possibility of their occurrence. Therefore, a certain quantitative measure is needed to compare events.
A quantitative measure of the possibility of an event occurring is probability ... The most widespread are two definitions of the probability of an event: classical and statistical.
Classic definition probability is associated with the concept of a favorable outcome. Let's dwell on this in more detail.
Let the outcomes of some test form a complete group of events and are equally possible, i.e. are the only possible, incompatible and equally possible. Such outcomes are called elementary outcomes, or cases... At the same time, they say that the test is reduced to case diagram or " urn scheme", Because any probabilistic problem for such a test can be replaced by an equivalent problem with urns and balls of different colors.
Exodus is called favorable event A if the occurrence of this event entails the occurrence of an event A.
According to the classical definition probability of 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 an event A; m- the number of cases favorable to the event A; n- the total number of cases.
Example 1.1. When throwing a dice, six outcomes are possible - 1, 2, 3, 4, 5, 6 points are dropped. What is the probability that an even number of points will appear?
Solution. Everything n= 6 outcomes form a complete group of events and are equally possible, i.e. are the only possible, incompatible and equally possible. Event A - “the appearance of an even number of points” - 3 outcomes (cases) are favored - 2, 4 or 6 points are dropped. According to 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 mentioned earlier, the classical definition of probability is applicable only to those events that can appear as a result of trials with the symmetry of possible outcomes, i.e. reduced to the scheme of cases. However, there is a large class of events, the probabilities of which 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 the coat of arms” and “appearance of tails” cannot be considered equally possible. Therefore, the formula for determining the probability according to the classical scheme is inapplicable in this case.
However, there is a different approach to assessing the likelihood of events based on how often the event will occur in the tests performed. In this case, a statistical definition of probability is used.
Statistical probabilityevent A is called the relative frequency (frequency) of the occurrence of this event in n tests performed, i.e.
 ,
| (1.2) |
where P * (A)- statistical probability of an event A; w (A)- relative frequency of the event A; m- the number of trials in which the event occurred A; n- the total number of tests.
Unlike mathematical probability P (A) considered in the classical definition, the statistical probability P * (A) is a characteristic experienced, experimental... In other words, the statistical probability of the event A is the number relative to 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 a shooter is said to hit 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 number 0.95, which is an indicator of the skill of the shooter, is usually very stable, i.e. the percentage of hits in most shooting will be almost the same for a given shooter, only in rare cases deviating somewhat significantly from its average value.
Another drawback of the classical definition of probability ( 1.1 ), limiting its use, is that it assumes a finite number of possible trial outcomes. In some cases, this disadvantage can be overcome by using a geometric definition of probability, i.e. finding the probability of a point hitting a certain area (segment, part of a plane, etc.).
Let a flat figure g forms part of a flat figure G(fig. 1.1). On the figure G a point is thrown at random. This means that all points of the region G"Equal" in relation to hitting it with a thrown random point. Assuming that the probability of an event A- hitting the thrown point on the figure g- proportional to the area of this figure and does not depend on its location relative to G nor from the form g, find
