EntranceGeometric and statistical determination of the probability of a random event. Probability of event
Kendall's rank correlation indicator, testing the corresponding hypothesis about the significance of the relationship.
2.Classical definition of probability. Properties 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 indicate weaknesses this definition and give other definitions that allow us to overcome the shortcomings of the classical definition.
Let's look at an example. Let an urn contain 6 identical, thoroughly mixed balls, 2 of them 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. Can this opportunity be quantified? It turns out that it is possible. This number is called the probability of an event (the appearance of a colored ball). Thus, probability is a number that characterizes the degree of possibility of an event occurring.
Let us set ourselves the task of giving a quantitative assessment of the possibility that a ball taken at random is colored. The appearance of a colored ball will be considered as event A. Each of the possible results of the test (the test consists of removing the ball from the urn) will be called elementary outcome (elementary event). We denote elementary outcomes by w 1, w 2, w 3, etc. In our example, the following 6 elementary outcomes are possible: w 1 - a white ball appears; w 2, w 3 - a red ball appeared; w 4, w 5, w 6 - a blue ball appears. It is easy to see that these outcomes form a complete group in pairs incompatible events(only one ball will appear) and they are equally possible (the ball is taken out at random, the balls are identical and thoroughly mixed).
We will call those elementary outcomes in which the event of interest to us occurs favorable this event. In our example, the following 5 outcomes favor event A (the appearance of a colored ball): w 2, w 3, w 4, w 5, w 6.
Thus, event A is observed if one of the elementary outcomes favoring A occurs in the test, no matter which one; in our example, A is observed if w 2, or w 3, or w 4, or w 5, or w 6 occurs. In this sense, event A is divided into several elementary events (w 2, w 3, w 4, w 5, w 6); an elementary event is not subdivided into other events. This is the difference between event A and an elementary event (an elementary outcome).
The ratio of the number of elementary outcomes favorable to event A to their total number is called the probability of event A and is denoted by P (A). In the example under consideration, there are 6 elementary outcomes; 5 of them favor event A. Consequently, the probability that the taken ball will be colored is equal to P (A) = 5 / 6. This number gives the quantitative assessment of the degree of possibility of the appearance of a colored ball that we wanted to find. Let us now give the definition of probability.
Probability of event A they call 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 event A is determined by the formula
where m is the number of elementary outcomes favorable to A; n is 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:
With in about s t in about 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 (A) = m / n = n / n = 1.
S in about with t in about 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 (A) = m / n = 0 / n = 0.
With in about with t in about 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 0< m < n, значит, 0 < m / n < 1, следовательно,
0 < Р (А) < 1
So, the probability of any event satisfies the double inequality
Remark: Modern rigorous courses in probability theory are built on a set-theoretic basis. Let us limit ourselves to presenting in the language of set theory the concepts discussed above.
Let one and only one of the events w i, (i = 1, 2, ..., n) occur as a result of the test. Events w i are called elementary events (elementary outcomes). It already follows from this that elementary events are pairwise incompatible. The set of all elementary events that can occur in a test is called space of elementary events W, and the elementary events themselves are points of space W.
Event A is identified with a subset (of space W), the elements of which are elementary outcomes favorable to A; event B is a subset of W whose elements are outcomes favorable to B, etc. Thus, the set of all events that can occur in a test is the set of all subsets of W. W itself occurs for any outcome of the test, therefore W is a reliable event; an empty subset of the space W is an impossible event (it does not occur under any outcome of the test).
Note that elementary events are distinguished from all events by the fact that each of them contains only one element W.
Each elementary outcome w i is assigned a positive number p i is the probability of this outcome, and
By definition, the probability P(A) of event A is equal to the sum of the probabilities of elementary outcomes favorable to A. From here it is easy to obtain that the probability of a reliable event is equal to one, an impossible event is equal to 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/n. Let event A be favored by m outcomes. The probability of event A is equal to the sum of the probabilities of outcomes favoring A:
P (A) = 1 / n + 1 / n + .. + 1 / n.
Considering that the number of terms is equal to m, we have
P(A) = m/n.
A classical definition of probability is obtained.
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. Each event A is associated with a non-negative real number R(A). This number is called the probability of 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.
3.Static determination of probability, relative frequency.
The classical definition does not require experimentation. While real applied problems have an infinite number of outcomes, and the classical definition in this case cannot provide an answer. Therefore, in such problems we will use static determination of probabilities, which is calculated after an experiment or experiment.
Static probability w(A) or relative frequency is the ratio of the number of favorable this event outcomes to the total number of actual tests performed.
w(A)=nm
The relative frequency of an event has stability property:
lim n→∞P(∣ ∣ nm−p∣ ∣ <ε)=1 (свойство устойчивости относительной частоты)
4. Geometric probabilities.
At geometric approach to the definition probabilities an arbitrary set is considered as the space of elementary events finite Lebesgue measure on a line, plane or space. Events are called all kinds of measurable subsets of the set.
Probability of event A is determined by the formula
where denotes Lebesgue measure of set A. With this definition of events and probabilities, everything A.N. Kolmogorov’s axioms are satisfied.
In specific tasks that boil down to the above probabilistic scheme, the test is interpreted as a random selection of a point in some area, and the event A– how the selected point hits a certain subregion A of the region. In this case, it is required that all points in the region have equal opportunity to be selected. This requirement is usually expressed in words “at random”, “randomly”, etc.
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 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 was said 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 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, when shooting is repeated many times 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
As mentioned above, the classical definition of probability assumes that all elementary outcomes are equally possible. The equality of experimental outcomes is concluded due to considerations of symmetry. Problems in which symmetry considerations can be used are rare in practice. In many cases it is difficult to provide reasons for believing that all elementary outcomes are equally possible. In this regard, it became necessary to introduce another definition of probability, called statistical. Let us first introduce the concept of relative frequency.
Relative frequency of the event, or frequency, is the ratio of the number of experiments in which this event occurred to the number of all experiments performed. Let us denote the frequency of the event A through W(A), Then
Where n– total number of experiments; m– number of experiments in which the event occurred A.
With a small number of experiments, the frequency of the event is largely random and can vary noticeably from one group of experiments to another. For example, with some ten coin tosses it is quite possible that the coat of arms will appear 2 times (frequency 0.2), with another ten tosses we may well get 8 coats of arms (frequency 0.8). However, with an increase in the number of experiments, the frequency of the event increasingly loses its random character; the random circumstances inherent in each individual experience cancel out in the mass, and the frequency tends to stabilize, approaching with minor fluctuations a certain average constant value. This constant, which is an objective numerical characteristic of a phenomenon, is considered the probability of a given event.
Statistical definition of probability: probability events is the number around which the frequency values of a given event are grouped in different series of a large number of tests.
The property of frequency stability, repeatedly tested experimentally and confirmed by the experience of mankind, is one of the most characteristic patterns observed in random phenomena. There is a deep connection between the frequency of an event and its probability, which can be expressed as follows: when we assess the degree of possibility of an event, we associate this assessment with a greater or lesser frequency of occurrence of similar events in practice.
Geometric probability
The classical definition of probability assumes that the number of elementary outcomes is finite. In practice, there are experiments for which the set of such outcomes is infinite. In order to overcome this drawback of the classical definition of probability, which is that it is not applicable to tests with an infinite number of outcomes, they introduce geometric probabilities – the probabilities of a point falling into an area.
Let us assume that a quadratable region is given on the plane G, i.e. area having area S G. In the area G contains area g area Sg. To the region G A dot is thrown at random. We will assume that the thrown point can fall into some part of the area G with a probability proportional to the area of this part and independent of its shape and location. Let the event A– “the thrown point hits the area g", then the geometric probability of this event is determined by the formula:
In the general case, the concept of geometric probability is introduced as follows. Let us denote the measure of the area g(length, area, volume) through mes g, and the measure of the area G- through mes G ; let also A– event “a thrown point hits the area g, which is contained in the area G" Probability of hitting the area g points thrown into the area G, is determined by the formula
.
Task. A square is inscribed in a circle. A dot is thrown into the circle at random. What is the probability that the point will fall into the square?
Solution. Let the radius of the circle be R, then the area of the circle is . The diagonal of the square is , then the side of the square is , and the area of the square is . The probability of the desired event is defined as the ratio of the area of the square to the area of the circle, i.e. 
.
Security questions
1. What is called a test (experience)?
2. What is an event?
3. What event is called a) reliable? b) random? c) impossible?
4. What events are called a) incompatible? b) joint?
5. What events are called opposite? Are they a) incompatible b) compatible or random?
6. What is called a complete group of random events?
7. If the events cannot all happen together as a result of the test, will they be pairwise incompatible?
8. Do events form A and the full group?
9. What elementary outcomes favor this event?
10. What definition of probability is called classical?
11. What are the limits of the probability of any event?
12. Under what conditions is classical probability applied?
13. Under what conditions is geometric probability applied?
14. What definition of probability is called geometric?
15. What is the frequency of an event?
16. What definition of probability is called statistical?
Test tasks
1. One letter is selected at random from the letters of the word “conservatory”. Find the probability that this letter is a vowel. Find the probability that it is the letter "o".
2. The letters “o”, “p”, “s”, “t” are written on identical cards. Find the probability that the word “cable” will appear on cards placed randomly in a row.
3. There are 4 women and 3 men in the team. 4 tickets to the theater are raffled off among brigade members. What is the probability that among the ticket holders there will be 2 women and 2 men?
4. Two dice are tossed. Find the probability that the sum of points on both dice is greater than 6.
5. The letters l, m, o, o, t are written on five identical cards. What is the probability that, taking out the cards one at a time, we will get the word “hammer” in the order they appeared?
6. Out of 10 tickets, 2 are winning. What is the probability that among five tickets taken at random, one is winning?
7. What is the probability that in a randomly chosen two-digit number the digits are such that their product is equal to zero.
8. A number not exceeding 30 is chosen at random. Find the probability that this number is a divisor of 30.
9. A number not exceeding 30 is chosen at random. Find the probability that this number is a multiple of 3.
10. A number not exceeding 50 is chosen at random. Find the probability that this number is prime.
In order to quantitatively compare events with each other according to the degree of their possibility, obviously, it is necessary to associate a certain number with each event, which is greater, the more possible the event. We will call this number the probability of an event. Thus, probability of an event is a numerical measure of the degree of objective possibility of this event.
The first definition of probability should be considered the classical one, which arose from the analysis of gambling and was initially applied intuitively.
The classical method of determining probability is based on the concept of equally possible and incompatible events, which are the outcomes of a given experience and form a complete group of incompatible events.
The simplest example of equally possible and incompatible events forming a complete group is the appearance of one or another ball from an urn containing several balls of the same size, weight and other tangible characteristics, differing only in color, thoroughly mixed before being removed.
Therefore, a test whose outcomes form a complete group of incompatible and equally possible events is said to be reducible to a pattern of urns, or a pattern of cases, or fits into the classical pattern.
Equally possible and incompatible events that make up a complete group will be called simply cases or chances. Moreover, in each experiment, along with cases, more complex events can occur.
Example: When throwing a dice, along with the cases A i - the loss of i-points on the upper side, we can consider such events as B - the loss of an even number of points, C - the loss of a number of points that are a multiple of three...
In relation to each event that can occur during the experiment, cases are divided into favorable, in which this event occurs, and unfavorable, in which the event does not occur. In the previous example, event B is favored by cases A 2, A 4, A 6; event C - cases A 3, A 6.
Classical probability the occurrence of a certain event is called the ratio of the number of cases favorable to the occurrence of this event to the total number of equally possible, incompatible cases that make up the complete group in a given experiment:
Where P(A)- probability of occurrence of event A; m- the number of cases favorable to event A; n- total number of cases.
Examples:
1) (see example above) P(B)= , P(C) =.
2) The urn contains 9 red and 6 blue balls. Find the probability that one or two balls drawn at random will turn out to be red.
A- a red ball drawn at random:
m= 9, n= 9 + 6 = 15, P(A)=
B- two red balls drawn at random:
The following properties follow from the classical definition of probability (show yourself):
1) The probability of an impossible event is 0;
2) The probability of a reliable event is 1;
3) The probability of any event lies between 0 and 1;
4) The probability of an event opposite to event A,
The classic definition of probability assumes that the number of outcomes of a trial is finite. In practice, very often there are tests, the number of possible cases of which is infinite. In addition, the weakness of the classical definition is that very often it is impossible to represent the result of a test in the form of a set of elementary events. It is even more difficult to indicate the reasons for considering the elementary outcomes of a test to be equally possible. Usually, the equipossibility of elementary test outcomes is concluded from considerations of symmetry. However, such tasks are very rare in practice. For these reasons, along with the classical definition of probability, other definitions of probability are also used.
Statistical probability event A is the relative frequency of occurrence of this event in the tests performed:
where is the probability of occurrence of event A;
Relative frequency of occurrence of event A;
The number of trials in which event A appeared;
Total number of trials.
Unlike classical probability, statistical probability is a characteristic of experimental probability.
Example: To control the quality of products from a batch, 100 products were selected at random, among which 3 products turned out to be defective. Determine the probability of marriage.
The statistical method of determining probability is applicable only to those events that have the following properties:
The events under consideration should be the outcomes of only those tests that can be reproduced an unlimited number of times under the same set of conditions.
Events must have statistical stability (or stability of relative frequencies). This means that in different series of tests the relative frequency of the event changes little.
The number of trials resulting in event A must be quite large.
It is easy to verify that the properties of probability arising from the classical definition are also preserved in the statistical definition of probability.
Probability is the degree (measure, quantitative assessment) of the possibility of the occurrence of some event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - incredible or unlikely. The preponderance of positive reasons over negative ones, and vice versa, can be to varying degrees, as a result of which the probability (and improbability) can be greater or lesser. Therefore, probability is often assessed at a qualitative level, especially in cases where a more or less accurate quantitative assessment is impossible or extremely difficult. Various gradations of “levels” of probability are possible.
The classic definition of probability is based on the concept of equal probability of outcomes. The probability is the ratio of the number of outcomes favorable for a given event to the total number of equally possible outcomes. For example, the probability of getting heads or tails in a random coin toss is 1/2 if it is assumed that only these two possibilities occur and that they are equally possible. This classical “definition” of probability can be generalized to the case of an infinite number of possible values - for example, if some event can occur with equal probability at any point (the number of points is infinite) of some limited region of space (plane), then the probability that it will occur in of some part of this feasible region is equal to the ratio of the volume (area) of this part to the volume (area) of the region of all possible points.
The probabilistic description of certain phenomena has become widespread in modern science, in particular in econometrics, statistical physics of macroscopic (thermodynamic) systems, where even in the case of a classical deterministic description of the movement of particles, a deterministic description of the entire system of particles does not seem practically possible or appropriate. In quantum physics, the processes described are themselves probabilistic in nature.
The emergence of the concept and theory of probability
The first works on the doctrine of probability date back to the 17th century. Such as the correspondence of the French scientists B. Pascal, P. Fermat (1654) and the Dutch scientist H. Huygens (1657), who gave the earliest known scientific interpretation of probability]. Essentially, Huygens already operated with the concept of mathematical expectation. Swiss mathematician J. Bernoulli established the law of large numbers for the design of independent trials with two outcomes (posthumously, 1713). In the 18th century - beginning of the 19th century. probability theory is developed in the works of A. Moivre (England) (1718), P. Laplace (France), C. Gauss (Germany) and S. Poisson (France). The theory of probability begins to be applied in the theory of observation errors, which developed in connection with the needs of geodesy and astronomy, and in the theory of shooting. It should be noted that the law of error distribution was essentially proposed by Laplace, first as an exponential dependence on the error without taking into account the sign (in 1774), then as an exponential function of the squared error (in 1778). The latter law is usually called the Gaussian distribution or normal distribution. Bernoulli (1778) introduced the principle of the product of the probabilities of simultaneous events. Adrien Marie Legendre (1805) developed the method of least squares.
In the second half of the 19th century. The development of probability theory is associated with the work of Russian mathematicians P. L. Chebyshev, A. M. Lyapunov and A. A. Markov (senior), as well as work on mathematical statistics by A. Quetelet (Belgium) and F. Galton (England) and statistical physicist L. Boltzmann (in Austria), who created the basis for a significant expansion of the problems of probability theory. The currently most common logical (axiomatic) scheme for constructing the foundations of probability theory was developed in 1933 by the Soviet mathematician A. N. Kolmogorov.
Classic definition of probability:
According to the classical definition, the probability of a random event P(A) is equal to the ratio of the number of outcomes favorable to A to the total number of outcomes that make up the space of elementary events, i.e.
probability static classical theory
Calculating probabilities in this case comes down to counting the elements of a particular set and often turns out to be a purely combinatorial task, sometimes very difficult.
The classical definition is justified when it is possible to predict the probability based on the symmetry of the conditions under which the experiment takes place, and therefore the symmetry of the outcomes of the test, which leads to the concept of “equal possibility” of outcomes.
For example. If a geometrically regular die made of a homogeneous material is thrown so that it manages to make a sufficiently large number of revolutions before falling, then the loss of any of its faces is considered an equally possible outcome.
For the same reasons of symmetry, the outcomes of such an experiment as removing thoroughly mixed and indistinguishable to the touch white and black balls are considered equally possible, so that after registering the color, each ball is returned back to the vessel and, after thorough mixing, the next ball is removed.
Most often, such symmetry is observed in artificially organized experiments, such as gambling.
Thus, the classical definition of probability is associated with the concept of equal opportunity and is used for experiments that reduce to a case scheme. To do this, it is necessary that the events e1, e2, en be incompatible, that is, no two of them can appear together; such that they form a complete group, i.e. they exhaust all possible outcomes (it cannot be that as a result of experience none of them occurred); equally possible, provided that the experiment provides the same possibility of the appearance of each of them.
Not every experiment satisfies the case scheme. If the symmetry condition is violated, then there is no scheme of cases.
Formula (1.1), the "classical formula", has been used to calculate the probabilities of events from the very beginning of the emergence of the science of random phenomena.
Those experiments that did not have symmetry were “adjusted” to fit the scheme of cases. Currently, along with the “classical formula,” there are methods for calculating probabilities when the experiment is not reduced to a scheme of cases. For this purpose, the statistical definition of probability is used.
The concept of statistical probability will be introduced later, but now let’s return to the classical formula.
Consider the following examples.
Example 1. The experiment consists of tossing two coins. Find the probability that at least one coat of arms will appear.
Solution. Random event A - the appearance of at least one coat of arms.
The space of elementary events in this experiment is determined by the following outcomes: E = (GG, GR, RG, RR), which are respectively designated e1, e2, e3, e4. Thus,
E=e1, e2, e3, e4; n=4.
It is necessary to determine the number of outcomes from E that favor the emergence of A. These are e1, e2, e3; their number is m=3.
Using the classical formula for determining the probability of event A, we have
Example 2. There are 3 white and 4 black balls in an urn. One ball is drawn from the urn. Find the probability that this ball is white.
Solution. Random event A - the appearance of a white ball. The space of elementary events E includes outcomes e1, e2, e3, e4, e5, e6, e7, where ei is the appearance of one ball (white or black);
E=(e1, e2, e3, e4, 5, e6, e7), n=7.
A random event A in space E is favored by 3 outcomes; m=3. Hence,
Example 3. There are 3 white and 4 black balls in an urn. Two balls are drawn from the urn. Find the probability that both will be white.
Solution. Random event A - both balls will be white.
Example 3 differs from example 2 in that in example 3 the outcomes that make up the space of elementary outcomes E will not be individual balls, but combinations of 7 balls by 2. That is, in order to determine the dimension of E, it is necessary to determine the number of combinations of 7 by 2. To do this, you need to use combinatorics formulas, which are given in the “Combinatorial method” section. In this case, to determine the number of combinations from 7 to 2, the formula is used to determine the number of combinations
since the choice is made without returning and the order in which the balls appear is unimportant. Thus,
The number of combinations favorable for the occurrence of event A is defined as
Hence, .
Statistical definition of probability
When looking at the results of individual tests, it is very difficult to find any patterns. However, in a sequence of identical tests, it is possible to detect the stability of some average characteristics. The frequency of any event in a given series of n trials is the ratio m/n, the number m of those trials in which event A occurred, to the total number of trials n. In almost every sufficiently long series of tests, the frequency of event A is set around a certain value, which is taken as the probability of event A. The stability of the frequency value is confirmed by special experiments. Statistical patterns of this kind were first discovered using the example of gambling, that is, using the example of those tests that are characterized by the possibility of outcomes. This opened the way for a statistical approach to the numerical determination of the probability when the symmetry condition of the experiment is violated. The frequency of event A is called statistical probability, which is denoted
where mA is the number of experiments in which event A appeared;
n is the total number of experiments.
Formulas (1.1) and (1.2) for determining probability are superficially similar, but they are essentially different. Formula (1.1) serves to theoretically calculate the probability of an event under given experimental conditions. Formula (1.2) serves to experimentally determine the frequency of an event. To use formula (1.2), experienced statistical material is needed.
Axiomatic approach to determining probability
The third approach to determining probability is the axiomatic approach, in which probabilities are specified by listing their properties.
The accepted axiomatic definition of probability was formulated in 1933 by A. N. Kolmogorov. In this case, the probability is specified as a numerical function P(A) on the set of all events determined by a given experiment, which satisfies the following axioms:
P(A)=1, if A is a reliable event.
If A and B are inconsistent.
Basic properties of probability
For each random event A its probability is determined, and.
For a reliable event U, the equality P(U)=1 holds. Properties 1 and 2 follow from the definition of probability.
If events A and B are incompatible, then the probability of the sum of events is equal to the sum of their probabilities. This property is called the formula for adding probabilities in a particular case (for incompatible events).
For arbitrary events A and B
This property is called the formula for adding probabilities in the general case.
For opposite events A, equality holds.
In addition, an impossible event is introduced, designated, which is not promoted by any outcome from the space of elementary events. The probability of an impossible event is 0, P()=0.
Example. The probability that a family randomly selected as a result of a survey has color, black-and-white, or color and black-and-white televisions is 0.86, respectively; 0.35; 0.29. What is the probability that a family has a color or black and white TV?
Solution. Let event A be that the family has a color TV.
Event B is that the family has a black and white TV.
Event C is that the family has either a color or black and white television. Event C is defined through A and B in the form, A and B are compatible, therefore
Combinatorial method
In many probability problems, it is necessary to list all the possible outcomes of an experiment or elementary events that are possible in a given situation, or to calculate their number. To do this, you can use the following rules.
Rule 1. If an operation consists of two steps, in which the first can be done in n1 ways and the second can be done in n2 ways, then the entire operation can be done in n1·n2 ways.
The word "operation" refers to any procedure, process or method of choice.
To confirm this rule, consider an operation that consists of steps xi and yi, step x can be carried out in n1 ways, i.e. , step y can be carried out in n2 ways, i.e. , then the series of all possible ways can be represented by the following n1n2 pairs:
Example. How many possible outcomes are there in an experiment that involves tossing two dice?
Solution. By x and y in this case we mean the loss of any face on the first die and on the second die. The loss of a face on the first die is possible in six ways xi, ; The face of the second die can also fall out in six ways xj, .
Total possible ways 6.6=36.
Rule 2. If an operation consists of k steps, in which the first can be done in n1 ways, the second in n2 ways, the third in n2 ways, etc., k-th ways, then the entire operation can be done in n1·n2…nk steps .
Example. The quality inspector wants to select a part from each of four containers containing 4, 3, 5 and 4 parts respectively. In how many ways can he do this?
Solution. The total number of ways is determined as 4·3·5·4=240.
Example. In how many possible ways can a student answer a test of 20 questions if he can answer “yes” or “no” to each question?
Solution. All possible ways 2·2...2=220=1048576.
Often in practice a situation arises when objects must be ordered.
For example: in how many different ways can 6 people sit around a table? Their different arrangements are called permutations.
Example. How many permutations are possible for the letters a, b, c?
Solution. Possible locations abc, acb, bac, bca, cab, cba. The number of possible locations is six.
Generalizing this example, for n objects there are only n·(n-1)(n-2)…3 ·2 ·1 different ways or n!, i.e. the number of permutations n!=1·2·3...· (n-2)(n-1)n, with 0!=1.
Rule 3. The number of permutations of n different objects is equal to n!.
Example. The number of permutations of four letters is 4!=24, but what number of permutations will be obtained if you choose 2 letters out of four?
Solution. We have to fill in two four letter positions. For the first position - 4 ways, for the second position - 3 ways. Therefore, using rule 1, we have 4·3=12.
Generalizing this example to n different objects, from which r objects are selected without returning for r > 0, in total there are n(n-1)...(n-r+1). We denote this number, and the resulting combinations are called placements.
Rule 4. The number of placements of n objects by r is defined as
(for r = 0,1,...,n).
Permutations where objects are arranged in a circle are called circular permutations. Two circular permutations are not different (but count as only one) if the corresponding objects in the two arrangements have the same objects on the left and right.
For example: if four people are playing bridge, we will not get different arrangements if all players move one chair to the right.
Example. How many circular permutations are possible from four people playing bridge? Solution. If we arbitrarily take the position of one of the four players as fixed, we can position the other three players 3! ways, in other words, we have six different circular permutations.
Generalizing this example, we get the following rule.
Rule 5. The number of permutations of n different objects located in a circle is (n-1)!.
So far it has been assumed that the n objects from which we select r objects and form the permutations are distinct. Thus, the formulas mentioned earlier cannot be used to determine the number of ways the letters in the word "book" can be arranged, or the number of ways three copies of one novella and one copy of each of the other four novellas can be arranged on a shelf.
Example. How many different permutations of letters are there in the word "book"?
Solution. If it is important to distinguish the letters O, then we denote them O1, O2 and then we will have 4!=24 different permutations of letters in O1, O2 and K. However, if we omit the indices, then O1 O2 and O2, O1 are no longer distinguished, then the total number permutations are equal.
Example. How many different ways can three copies of one novella and one copy of the other four novellas be arranged on a shelf?
Solution. If we designate three copies of the first novella as a1, a2, a3 and the other four novellas - b, c, d and e, then in this case we have 7! different ways and 3! way to arrange a1, a2, a3.
If you omit the indices, then there are different ways to arrange copies.
Summarizing these arguments, we obtain the following rule.
Rule 6. The number of permutations of n objects in which n1 are of one kind, n2 are of the second kind, ..., nk are of the kth kind and n1+n2+...+nk=n,
There are many problems in which you need to determine the number of ways to select r objects from n different objects, regardless of the order in which they are selected. Such combinations are called combinations.
Example. In how many ways can three candidates be selected from 20 people for a public poll?
Solution. If order is important to us when selecting candidates, then the number of combinations, but each row of three candidates can be selected 3! In ways; if the order of selection is not important, then the total selection methods.
Combinations without returning r objects from n different objects that differ in the objects themselves, but not in their order, are called combinations.
Rule 7. The number of combinations of r objects from n different objects is determined by the number, the number of combinations can be denoted as.
Example. In how many different ways can you get 2 heads and 4 tails with six coin tosses?
Solution. Since the order in which heads and tails are obtained is not important, then, applying rule 7, we get.
Example. How many different committees of two chemists and one physicist can be formed on the faculty of a small college having 4 chemists and 3 physicists.
Solution. The number of combinations of four chemists of 2 can be obtained in (six) ways.
One of the three physicists can be selected in (three) ways.
The number of committees, in accordance with rule 1, is determined as 6·3=18.
Example. In how many ways can a row of four objects be divided into three rows containing two, one, and one objects, respectively?
Solution. Let us denote these four objects by the letters a, b, c, d. The number of splits into two, one and one will be 12:
A partition of two objects can be obtained in ways that give 6 possibilities. The number of ways to form the second partition. And for the third partition, the number of ways is 1.
According to rule 2, the total number of partitioning methods is (6·2·1)=12.
Generalizing this example, we obtain the following rule.
Rule 8. The number of ways in which a series of n different objects can be divided into k parts with n1 objects in the 1st part, n2 in the 2nd part, ... and nk in the kth is given by
Example. In how many ways can 7 businessmen be accommodated in one three-room and two two-room hotel rooms?
Solution. According to Rule 8, this can be done in (two hundred) ways.
Proof of Rule 8
Since n1 objects can be selected in a number of ways, n2 can be selected
According to rule 2, the total number of ways will be determined in the form
Assignment for independent work
1. Ten books are placed at random on one shelf. Determine the probability that three specific books will be nearby.
Answer: 0.066.
2. Three cards are drawn at random from a deck of cards (52 cards). Find the probability that it will be a three, a seven and an ace.
Answer: 0.0029.
3. There are five tickets worth 1 ruble each;
three tickets costing 3 rubles each;
two tickets cost 5 rubles each.
Three tickets are selected at random. Determine the probability that:
a) at least two of these tickets have the same price.
Answer: 0.75;
b) all three tickets cost 7 rubles.
Answer: 0.29.
4. The wallet contains three coins of 20 kopecks and seven coins of 3 kopecks. One coin is taken at random, and then a second coin of 20 kopecks is taken out.
Determine the probability that the first coin also has a denomination of 20 kopecks.
Answer: 0.22.
- 5. Out of ten lottery tickets, two are winning. Determine the probability that among five tickets taken at random:
- a) one winning one;
- b) two winning ones;
- c) at least one winning one.
Answer: 0.55, 0.22, 0.78.
6. There are n balls in the basket with numbers from 1 to n, the balls are removed at random one at a time without returning. What is the probability that during the first k draws, the numbers of the balls will coincide with the numbers of draws.
Answer: (n - k)!/n!
Literature used
- 1. http://kurs.ido.tpu.ru/courses/theory_ver/tema2/tema2.html
- 2. http://free.megacampus.ru/xbookM0018/index.html?go=part-003*page.htm
- 3. http://www.testent.ru/publ/studenty/vysshaja_matematika/klassicheskoe_opredelenie_verojatnosti/35-1-0-1121
- 4. http://ru.wikipedia.org/
- 5. http://www.kolasc.net.ru/cdo/books/tv/page15.html