Through which financial planning is carried out. Financial planning
Let's analyze the classical definition of probability using formulas and examples.
Random events are called incompatible if they cannot occur at the same time. For example, when we toss a coin, one thing will fall out - a "coat of arms" or a number "and they cannot appear at the same time, since it is logical that this is impossible. Events such as hit and miss after a shot is fired can be incompatible.
Random events of a finite set form full group pairwise incompatible events, if at each trial one appears, and only one of these events is the only possible one.
Consider the same coin tossing example:
First coin Second coin Events
1) "coat of arms" "coat of arms"
2) "coat of arms" "number"
3) "number" "coat of arms"
4) "number" "number"
Or abbreviated - "YY", - "MS", - "CH", - "CH".
The events are called equally possible, if the conditions of the study provide the same possibility of the appearance of each of them.
As you understand, when you toss a symmetrical coin, then it has the same possibilities, and there is a chance that both the “coat of arms” and the “number” will fall out. The same applies to throwing a symmetrical dice, since there is a possibility that faces with any number of 1, 2, 3, 4, 5, 6 may appear.
Let's say that now we throw the cube with a shift in the center of gravity, for example, towards the side with the number 1, then the opposite side, that is, the side with a different number, will most often fall out. Thus, in this model, the occurrence possibilities for each of the digits from 1 to 6 will be different.
Equally possible and uniquely possible random events are called cases.
There are random events that are cases, and there are random events that are not cases. Below are examples of these events.
Those cases, as a result of which a random event appears, are called favorable cases for this event.
If we denote by , which affect the event in all possible cases, and through - the probability of a random event , then we can write down the well-known classical definition of probability:
Definition
The probability of an event is the ratio of the number of cases favorable to this event to the total number of all possible cases, that is:
Probability Properties
The classical probability has been considered, and now we will analyze the main and important properties of probability.
Property 1. The probability of a certain event is equal to one.
For example, if all the balls in the bucket are white, then the event , randomly select a white ball, is affected by the cases, .
Property 2. The probability of an impossible event is zero.
Property 3. The probability of a random event is a positive number:
Hence, the probability of any event satisfies the inequality:
Now let's solve some examples on the classical definition of probability.
Examples of the classical definition of probability
Example 1
A task
There are 20 balls in a basket, 10 of them are white, 7 are red and 3 are black. One ball is chosen at random. A white ball (event ), a red ball (event ), and a black ball (event ) are selected. Find the probability of random events.
Solution
According to the condition of the problem, contribute to , and cases of possible, therefore, according to the formula (1):
is the probability of a white ball.
Similarly for red:
And for black: .
Answer
The probability of a random event , , .
Example 2
A task
There are 25 identical electric lamps in a box, 2 of them are defective. Find the probability that a randomly selected light bulb is not defective.
Solution
According to the condition of the problem, all lamps are the same and only one is selected. Total possibilities to choose . Among all 25 lamps, two are defective, which means that the remaining lamps are suitable. Therefore, according to formula (1), the probability of choosing a suitable electric lamp (event ) is equal to:
Answer
The probability that a randomly selected light bulb is not defective = .
Example 3
A task
Two coins are tossed at random. Find the probability of such events:
1) - on both coins the coat of arms fell out;
2) - on one of the coins a coat of arms fell out, and on the second - a number;
3) - numbers fell out on both coins;
4) - at least once the coat of arms fell out.
Solution
Here we are dealing with four events. Let us establish which cases contribute to each of them. The event is facilitated by one case, this is when the coat of arms fell out on both coins (abbreviated as “GG”).
To deal with the event, imagine that one coin is silver and the second is copper. When tossing coins, there may be cases:
1) on a silver coat of arms, on a copper coat of arms - a number (let's denote it as “MS”);
2) on a silver number, on a copper one - a coat of arms (- "ChG").
Hence, the events are facilitated by the cases and .
The event is facilitated by one case: numbers fell out on both coins - “CH”.
Thus, the events or (YY, MG, TY, FF) form a complete group of events, all of these events are incompatible, since only one of them occurs as a result of the toss. In addition, for symmetric coins, all four events are equally likely, so they can be considered cases. There are four possible events.
An event is facilitated by only one event, so its probability is:
Two cases contribute to the event, so:
The probability of an event is the same as for:
Three cases contribute to the event: YY, YY, YY and therefore:
Since the events GY, MS, CH, CH are considered, which are equally probable and create a complete group of events, then the appearance of any of them is a reliable event (we denote it by the letter , which is facilitated by all 4 cases. Therefore, the probability:
Hence, the first property of probability is confirmed.
Answer
Probability of an event.
Probability of an event.
Probability of an event.
Probability of an event.
Example 4
A task
Two dice with the same and regular geometric shape are thrown. Find the probability of all possible sums on both sides that fall out.
Solution
To make it easier to solve the problem, imagine that one cube is white and the other is black. With each of the six faces of the white die and one of the six faces of the black die can also fall, so there will be all possible pairs.
Since the possibility of the appearance of faces on a separate die is the same (cubes of the correct geometric shape!), Then the probability of the appearance of each pair of faces will be the same, moreover, as a result of tossing, only one of the pairs falls out. Event values are incompatible, unique. These are cases, and there are 36 possible cases.
Now consider the possibility of the value of the sum on the faces. Obviously, the smallest sum is 1 + 1 = 2, and the largest is 6 + 6 = 12. The rest of the sum increases by one, starting from the second. Let's denote the events whose indices are equal to the sum of the points that fell on the faces of the dice. For each of these events, we write favorable cases using the notation , where is the sum, are the points on the upper face of the white die, and are the points on the face of the black die.
So for an event:
for – one case (1 + 1);
for – two cases (1 + 2; 2 + 1);
for – three cases (1 + 3; 2 + 2; 3 + 1);
for – four cases (1 + 4; 2 + 3; 3 + 2; 4 + 1);
for – five cases (1 + 5; 2 + 4; 3 + 3; 4 + 2; 5 + 1);
for – six cases (1 + 6; 2 + 5; 3 + 4; 4 + 3; 5 + 2; 6 + 1);
for – five cases (2 + 6; 3 + 5; 4 + 4; 5 + 3; 6 + 2);
for – four cases (3 + 6; 4 + 5; 5 + 4; 6 + 3);
for – three cases (4 + 6; 5 + 5; 6 + 4);
for – two cases (5 + 6; 6 + 5);
for – one case (6 + 6).
So the probabilities are:
Answer
Example 5
A task
Before the festival, three participants were offered to draw lots: each of the participants in turn approaches the bucket and randomly chooses one of three cards with numbers 1, 2 and 3, which means the serial number of the performance of this participant.
Find the probability of such events:
1) - the serial number in the queue coincides with the card number, that is, the serial number of the performance;
2) - no number in the queue matches the performance number;
3) - only one of the numbers in the queue matches the performance number;
4) – at least one of the numbers in the queue matches the performance number.
Solution
The possible results of choosing cards are permutations of three elements, the number of such permutations is equal to . Each permutation is an event. Let's denote these events as . We assign the corresponding permutation to each event in parentheses:
; ; ; ; ; .
The listed events are equally possible and uniform, that is, these are the cases. Denote as follows: (1h, 2h, 3h) - the corresponding numbers in the queue.
Let's start with the event. Favorable is only one case, therefore:
Favorable for the event are two cases and , therefore:
The event is facilitated by 3 cases: , therefore:
In addition to , the event also contributes to , that is:
Answer
The probability of an event is .
The probability of an event is .
The probability of an event is .
The probability of an event is .
The classical definition of probability - theory and problem solving updated: September 15, 2017 by: Scientific Articles.Ru
RUSSIAN ACADEMY OF THE NATIONAL ECONOMY AND PUBLIC SERVICE UNDER THE PRESIDENT OF THE RUSSIAN FEDERATION
OREL BRANCH
Department of Sociology and information technologies
Typical calculation No. 1
in the discipline "Probability Theory and Mathematical Statistics"
on the topic "Fundamentals of Probability Theory"
Eagle - 2016.
Objective: consolidation of theoretical knowledge on the topic of the foundations of the theory of probability, by solving typical problems. Mastering the concepts of the main types of random events and developing the skills of algebraic operations on events.
Job submission requirements: the work is done in handwritten form, the work must contain all the necessary explanations and conclusions, the formulas must contain a decoding of the accepted designations, the pages must be numbered.
Variant number corresponds to the student's serial number in the group list.
Basic theoretical information
Probability theory- a branch of mathematics that studies the patterns of random phenomena.
The concept of an event. Event classification.
One of the basic concepts of probability theory is the concept of an event. Events are indicated in capital Latin letters. BUT, IN, FROM,…
Event- this is a possible result (outcome) of a test or experience.
Testing is understood as any purposeful action.
Example : The shooter shoots at the target. A shot is a test, hitting a target is an event.
The event is called random , if under the conditions of a given experiment it can both occur and not occur.
Example : Shot from a gun - test
Inc. BUT- hitting the target
Inc. IN– miss – random events.
The event is called authentic if as a result of the test it must necessarily occur.
Example : Drop no more than 6 points when throwing a dice.
The event is called impossible if, under the conditions of the given experiment, it cannot occur at all.
Example : More than 6 points rolled when throwing a die.
The events are called incompatible if the occurrence of one of them precludes the occurrence of any other. Otherwise, the events are called joint.
Example : A dice is thrown. A roll of 5 eliminates a roll of 6. These are incompatible events. A student receiving “good” and “excellent” grades in exams in two different disciplines is a joint event.
Two incompatible events, of which one must necessarily occur, are called opposite . Event opposite to event BUT designate Ā .
Example : The appearance of the "coat of arms" and the appearance of "tails" when tossing a coin are opposite events.
Several events in this experience are called equally possible if there is reason to believe that none of these events is more possible than the others.
Example : drawing ace, tens, queens from a deck of cards - events are equally likely.
Several events form full group if, as a result of the test, one and only one of these events must necessarily occur.
Example : Dropping the number of points 1, 2, 3, 4, 5, 6 when throwing a die.
The classic definition of the probability of an event. Probability Properties
For practical activities it is important to be able to compare events according to the degree of possibility of their occurrence.
Probability An event is a numerical measure of the degree of objective possibility of an event occurring.
Let's call elementary outcome each of the equally likely test results.
Exodus is called favorable (favorable) event BUT, if its occurrence entails the occurrence of an event BUT.
Classic definition : event probability BUT is equal to the ratio of the number of outcomes favorable for a given event to the total number of possible outcomes.
(1)where P(A) is the probability of an event BUT,
m- the number of favorable outcomes,
n is the number of all possible outcomes.
Example : There are 1000 tickets in the lottery, of which 700 are not winning. What is the probability of winning on one purchased ticket.
Event BUT- purchased a winning ticket
Number of possible outcomes n=1000 is the total number of lottery tickets.
Number of outcomes favoring the event BUT is the number of winning tickets, i.e., m=1000-700=300.
According to the classical definition of probability: 
Answer:
.
Note event probability properties:
1) The probability of any event is between zero and one, i.e. 0≤ P(A)≤1.
2) The probability of a certain event is 1.
3) The probability of an impossible event is 0.
In addition to the classical, there are also geometric and statistical definitions of probability.
Elements of combinatorics.
Combinatorics formulas are widely used to calculate the number of outcomes favorable to the event in question or the total number of outcomes.
Let there be a set N from n various elements.
Definition 1: Combinations, each of which includes all n elements and which differ from each other only by the order of the elements are called permutations from n elements.
P n=n! (2), where n! (n-factorial) - product n the first numbers of the natural series, i.e.
n! = 1∙2∙3∙…∙(n–1)∙n
So, for example, 5!=1∙2∙3∙4∙5 = 120
Definition 2: m elements ( m≤n) and differing from each other either in the composition of the elements or their order are called placements from n on m elements.
(3) Definition 3: Combinations, each containing m elements ( m≤n) and differing from each other only in the composition of the elements are called combinations from n on m elements.
(4)
Comment: changing the order of elements within the same combination does not result in a new combination.
We formulate two important rules that are often used in solving combinatorial problems
Sum rule: if object BUT can be chosen m ways, and the object IN – n ways, then the choice is either BUT or IN can be done m+n ways.
Product rule: if object BUT can be chosen m ways, and the object IN after each such choice, one can choose n ways, then a pair of objects BUT And IN can be selected in that order. m ∙ n ways.
1. Statement of the main theorems and probability formulas: addition theorem, conditional probability, multiplication theorem, independence of events, total probability formula.
Goals: creation of favorable conditions for the introduction of the concept of the probability of an event; familiarity with the basic theorems and formulas of probability theory; enter the total probability formula.
Lesson progress:
Random experiment (experiment) is a process in which different outcomes are possible, and it is impossible to predict in advance what the result will be. The possible mutually exclusive outcomes of an experience are called its elementary events . The set of elementary events will be denoted by W.
random event an event is called, about which it is impossible to say in advance whether it will occur as a result of experience or not. Each random event A that occurred as a result of the experiment can be associated with a group of elementary events from W. The elementary events that make up this group are called favorable to the occurrence of event A.
The set W can also be considered as a random event. Since it includes all elementary events, it will necessarily occur as a result of experience. Such an event is called authentic .
If for a given event there are no favorable elementary events from W, 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 an equal opportunity 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 occur. The event opposite to event A is denoted by .
Events A and B are called incompatible if the occurrence of one of them excludes the occurrence of the other. Events A 1 , A 2 , ..., A n are called pairwise incompatible, if any two of them are incompatible. Events A 1 , A 2 , ..., An form complete system pairwise incompatible events if, as a result of the test, one and only one of them is sure to occur.
The sum (combination) of events A 1 , A 2 , ..., A n such an event C is called, which consists in the fact that at least one of the events A 1 , A 2 , ..., A n has occurred The sum of events is denoted as follows:
C \u003d A 1 + A 2 + ... + A n.
The product (intersection) of events A 1 , A 2 , ..., A n such an event P is called, which consists in the fact that all events A 1 , A 2 , ..., A n occurred simultaneously. The product of events is denoted
The probability P(A) in the theory of probability acts as a numerical characteristic of the degree of possibility of the occurrence of any particular random event A with repeated repetition of tests.
For example, in 1000 throws of a die, the number 4 comes up 160 times. The ratio 160/1000 = 0.16 shows the relative frequency of the number 4 falling out in this series of tests. More generally random event frequency And when conducting a series of experiments, they call the ratio of the number of experiments in which a given event occurred to the total number of experiments:
where P*(A) is the frequency of event A; m is the number of experiments in which event A occurred; n is the total number of experiments.
The probability of a random event A is called 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 do an unlimited number of tests in order to determine the probability. There is no need for this. 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 occur more often than others ( equally probable events), we can determine the probability based on theoretical considerations. For example, let's find out in the case of tossing a coin, the frequency of the coat of arms falling out (event A). Various experimenters have shown in several thousand trials that the relative frequency of such an event takes values close to 0.5. given that the appearance of the coat of arms and the opposite side of the coin (event B) are equally likely events if the coin is symmetrical, the judgment P(A)=P(B)=0.5 could be made without determining the frequency of these events. On the basis of the concept of "equal probability" of events, another definition of probability is formulated.
Let the event A under consideration occur in m cases, which are called favorable to A, and do 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 elementary events favorable to it to their total number(classical definition of the probability of an event):
where m is the number of elementary events that favor event A; n - The total number of elementary events.
Let's look at a few examples:
Example #1:An urn contains 40 balls: 10 black and 30 white. Find the probability that a randomly chosen ball is black.
The number of favorable cases is equal to the number of black balls in the urn: m = 10. The total number of equally probable events (taking out one ball) is equal to the total number of balls in the urn: n = 40. These events are incompatible, 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 die.
When throwing a die, six equally possible incompatible events are realized: the appearance of one digit: 1,2,3,4,5 or 6, i.e. n = 6. Favorable cases are the loss of one of the numbers 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.
Probability addition theorem: the probability of occurrence of one (no matter what) event from several incompatible events is equal to the sum of their probabilities.
For two incompatible events A and B, the probabilities of these events is equal to the sum of their probabilities:
P(A or B)=P(A) + P(B).
Example #3:Find the probability of getting 1 or 6 when throwing a dice.
Event A (roll 1) and B (roll 6) are equally likely: P(A) = P(B) = 1/6, so 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 incompatible events.
Example #4:An 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 a ball from the urn.
The probability of drawing a white ball (event A) is P(A) = 10/50 = 1/5, a black ball (event B) is P(B) = 20/50 = 2/5 and a red ball (event C) is P (C) = 5/50 = 1/10. From here, according to the formula for adding probabilities, we get P (A or B or C) \u003d P (A) + P (B) \u003d P (C) \u003d 1/5 + 2/5 + 1/10 \u003d 7/10
The sum of the probabilities of two opposite events, as follows from the probability addition theorem, is equal to one:
P(A) + P() = 1
In the above example, taking out the white, black and red balls will be the event A 1 , P(A 1) = 7/10. The opposite event of 1 is drawing the blue ball. Since there are 15 blue balls, and total amount 50 balls, then 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 general, the probability of the sum of two events A and B is calculated as
P (A + B) \u003d 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 occurred or not, and vice versa, the probability of occurrence of event B does not depend on whether event A 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: One urn contains 5 black and 10 white balls, the other 3 black and 17 white. Find the probability that the first time balls are drawn from each urn, both balls are black.
Solution: the probability of drawing 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) \u003d P (A) P (B) \u003d (1/3) (3/20) \u003d 3/60 \u003d 1/20.
In practice, the probability of an event B often depends on whether some other event A has occurred or not. In this case, one speaks of conditional probability , i.e. the probability of event B given that event A has occurred. The conditional probability is denoted by P(B/A).
Brief theory
For a quantitative comparison of events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event. The probability of a random event a number is called, which is an expression of a measure of the objective possibility of the occurrence of an event.
The values that determine how significant are the objective grounds for counting on the occurrence of an event are characterized by the probability of the event. It must be emphasized that probability is an objective quantity that exists independently of the cognizer and is conditioned by the totality of conditions that contribute to the occurrence of an event.
The explanations that we have given to the concept of probability are not a mathematical definition, since they do not define this concept quantitatively. There are several definitions of the probability of a random event that are widely used in solving specific problems (classical, geometric definition of probability, statistical, etc.).
The classical definition of the probability of an event reduces this concept to a more elementary concept of equally probable events, which is no longer subject to definition and is assumed to be intuitively clear. For example, if a dice is a homogeneous cube, then the fallout of any of the faces of this cube will be equally probable events.
Let a certain event be divided into equally probable cases, the sum of which gives the event. That is, the cases from , into which it breaks up, are called favorable for the event, since the appearance of one of them ensures the offensive.
The probability of an event will be denoted by the symbol .
The probability of an event is equal to the ratio of the number of cases favorable to it, out of the total number of unique, equally possible and incompatible cases, to the number, i.e.
This is the classical definition of probability. Thus, in order to find the probability of an event, it is necessary, after considering the various outcomes of the test, to find a set of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m that favor this event, and then perform the calculation according to the above formula.
The probability of an event equal to the ratio of the number of outcomes of experience favorable to the event to the total number of outcomes of experience is called classical probability random event.
The following properties of probability follow from the definition:
Property 1. The probability of a certain event is equal to one.
Property 2. The probability of an impossible event is zero.
Property 3. The probability of a random event is a positive number between zero and one.
Property 4. The probability of the occurrence of events that form a complete group is equal to one.
Property 5. The probability of the occurrence of the opposite event is defined in the same way as the probability of the occurrence of event A.
The number of occurrences that favor the occurrence of the opposite event. Hence, the probability of the opposite event occurring is equal to the difference between unity and the probability of the event A occurring:
An important advantage of the classical definition of the probability of an event is that with its help the probability of an event can be determined without resorting to experience, but on the basis of logical reasoning.
When a set of conditions is met, a certain event will definitely happen, and the impossible will definitely not happen. Among the events that, when a complex of conditions is created, may or may not occur, the appearance of some can be counted on with more reason, on the appearance of others with less reason. If, for example, there are more white balls in the urn than black ones, then there are more reasons to hope for the appearance of a white ball when taken out of the urn at random than for the appearance of a black ball.
Seen on the next page.
Problem solution example
Example 1
A box contains 8 white, 4 black and 7 red balls. 3 balls are drawn at random. Find the probabilities of the following events: - at least 1 red ball is drawn, - there are at least 2 balls of the same color, - there are at least 1 red and 1 white ball.
The solution of the problem
We find the total number of test outcomes as the number of combinations of 19 (8 + 4 + 7) elements of 3 each:
Find the probability of an event– drawn at least 1 red ball (1,2 or 3 red balls)
Required probability:
Let the event- there are at least 2 balls of the same color (2 or 3 white balls, 2 or 3 black balls and 2 or 3 red balls)
Number of outcomes favoring the event:
Required probability:
Let the event– there is at least one red and one white ball
(1 red, 1 white, 1 black or 1 red, 2 white or 2 red, 1 white)
Number of outcomes favoring the event:
Required probability:
Answer: P(A)=0.773;P(C)=0.7688; P(D)=0.6068
Example 2
Two dice are thrown. Find the probability that the sum of the points is at least 5.
Solution
Let the event be the sum of points not less than 5
Let's use the classical definition of probability:
Total number of possible trial outcomes
The number of trials that favor the event of interest to us
On the dropped face of the first dice, one point, two points ..., six points can appear. similarly, six outcomes are possible on the second die roll. Each of the outcomes of the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary outcomes of the test is equal to the number of placements with repetitions (selection with placements of 2 elements from a set of volume 6):
Find the probability of the opposite event - the sum of points is less than 5
The following combinations of dropped points will favor the event:
| № | 1st bone | 2nd bone | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 1 | 4 | 3 | 1 | 5 | 1 | 3 |
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Probability is one of the basic concepts of probability theory. There are several definitions of this concept. Let us give a definition that is called classical.
Probability event is the ratio of the number of elementary outcomes that favor a given event to the number of all equally possible outcomes of experience in which this event can appear.
The probability of an event A is denoted by P(A)(here R- the first letter of the French word probability- probability).
According to the definition
where is the number of elementary test outcomes favoring the appearance of the event ;
The total number of possible elemental outcomes of the trial.
This definition of probability is called classic. It arose on initial stage development of the theory of probability.
The number is often referred to as the relative frequency of occurrence of the event. BUT in experience.
The greater the probability of an event, the more often it occurs, and vice versa, the lower the probability of an event, the less often it occurs. When the probability of an event is close to one or equal to one, then it occurs in almost all trials. Such an event is said to be almost certain, i.e., that one can certainly count on its offensive.
Conversely, when the probability is zero or very small, then the event occurs extremely rarely; such an event is said to be almost impossible.
Sometimes the probability is expressed as a percentage: R(A) 100% is the average percentage of the number of occurrences of the event A.
Example 2.13. When dialing a phone number, the subscriber forgot one digit and dialed it at random. Find the probability that the desired digit is dialed.
Solution.
Denote by BUT event - "the required number is dialed".
The subscriber could dial any of the 10 digits, so the total number of possible elementary outcomes is 10. These outcomes are incompatible, equally possible and form a complete group. Favors the event BUT only one outcome (the required number is only one).
The desired probability is equal to the ratio of the number of outcomes that favor the event to the number of all elementary outcomes:
The classical probability formula provides a very simple way to calculate probabilities that does not require experimentation. However, the simplicity of this formula is very deceptive. The fact is that when using it, as a rule, two very difficult questions arise:
1. How to choose a system of outcomes of experience so that they are equally likely, and is it possible to do this at all?
2. How to find numbers m And n?
If multiple subjects are involved in an experiment, it is not always easy to see equally likely outcomes.
The great French philosopher and mathematician d'Alembert entered the history of probability theory with his famous mistake, the essence of which was that he incorrectly determined the equiprobability of outcomes in an experiment with only two coins!
Example 2.14. ( d'Alembert error). Two identical coins are tossed. What is the probability that they fall on the same side?
d'Alembert's solution.
Experience has three equally possible outcomes:
1. Both coins will fall on the "eagle";
2. Both coins will fall on "tails";
3. One of the coins will land on heads, the other on tails.
Correct solution.
Experience has four equally possible outcomes:
1. The first coin will fall on the "eagle", the second also on the "eagle";
2. The first coin will fall on "tails", the second will also fall on "tails";
3. The first coin will land on heads, and the second on tails;
4. The first coin will land on tails, and the second on heads.
Of these, two outcomes will be favorable for our event, so the desired probability is equal to .
d'Alembert made one of the most common mistakes made when calculating probability: he combined two elementary outcomes into one, thereby making it unequal in probability to the remaining outcomes of the experiment.
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