Tasks for the classical definition of probability.
Solution examples

In the third lesson, we will consider various problems related to the direct application of the classical definition of probability. To effectively study the materials of this article, I recommend that you familiarize yourself with the basic concepts probability theory and basics of combinatorics. The problem of the classical determination of probability with a probability tending to one will be present in your independent / control work on terver, so we are getting ready for serious work. What's so serious, you ask? ... just one primitive formula. I warn against frivolity - thematic tasks are quite diverse, and many of them can easily confuse. In this regard, in addition to working out the main lesson, try to study additional tasks on the topic that are in the piggy bank ready-made solutions in higher mathematics. Decision methods are decision methods, but “friends” still “need to be known by sight”, because even a rich imagination is limited and there are also enough typical tasks. Well, I will try to make out the maximum number of them in good quality.

Let's remember the classics of the genre:

The probability of an event occurring in some trial is equal to the ratio , where:

is the total number of all equally possible, elementary outcomes of this test, which form full group of events;

- amount elementary outcomes favoring the event.

And immediately an immediate pit stop. Do you understand the underlined terms? It means clear, not intuitive understanding. If not, then it is still better to return to the 1st article on probability theory and only then move on.

Please do not skip the first examples - in them I will repeat one fundamentally important point, and also tell you how to properly format a solution and in what ways it can be done:

Task 1

An urn contains 15 white, 5 red and 10 black balls. 1 ball is drawn at random, find the probability that it will be: a) white, b) red, c) black.

Decision: the most important prerequisite for using the classical definition of probability is the ability to calculate the total number of outcomes.

There are 15 + 5 + 10 = 30 balls in the urn, and obviously the following facts are true:

– extraction of any ball is equally possible (equal opportunity outcomes), while the outcomes elementary and form full group of events (i.e. as a result of the test, one of the 30 balls will definitely be removed).

Thus, the total number of outcomes:

Consider the following event: – a white ball will be drawn from the urn. This event is favored elementary outcomes, so by the classical definition:
is the probability that a white ball will be drawn from the urn.

Oddly enough, even in such a simple problem, one can make a serious inaccuracy, which I already focused on in the first article on probability theory. Where is the pitfall here? It is incorrect to argue here that "since half of the balls are white, then the probability of drawing a white ball» . The classic definition of probability is ELEMENTARY outcomes, and the fraction must be written!

With other points similarly, consider the following events:

- a red ball will be drawn from the urn;
- A black ball will be drawn from the urn.

The event is favored by 5 elementary outcomes, and the event is favored by 10 elementary outcomes. So the corresponding probabilities are:

A typical verification of many terver problems is done using theorems on the sum of probabilities of events forming a complete group. In our case, the events form a complete group, which means that the sum of the corresponding probabilities must necessarily be equal to one: .

Let's check if this is so: , which I wanted to make sure of.

Answer:

In principle, the answer can be written in more detail, but personally I’m used to putting only numbers there - for the reason that when you start “stamping” tasks in hundreds and thousands, you strive to minimize the solution entry. By the way, about brevity: in practice, a “high-speed” design option is common. solutions:

Total: 15 + 5 + 10 = 30 balls in the urn. According to the classical definition:
is the probability that a white ball will be drawn from the urn;
is the probability that a red ball will be drawn from the urn;
is the probability that a black ball will be drawn from the urn.

Answer:

However, if there are several points in the condition, then the solution is often more convenient to draw up in the first way, which takes a little more time, but then it “puts everything on the shelves” and makes it easier to navigate the task.

Warm up:

Task 2

The store received 30 refrigerators, five of which have a factory defect. One refrigerator is randomly selected. What is the probability that it will be defect free?

Choose the design option that suits you and check the template at the bottom of the page.

In the simplest examples, the number of common and the number of favorable outcomes lie on the surface, but in most cases you have to dig up the potatoes yourself. The canonical series of problems about the forgetful subscriber:

Task 3

When dialing a phone number, the subscriber forgot the last two digits, but remembers that one of them is zero, and the other is odd. Find the probability that he will dial the correct number.

Note : zero is an even number (divisible by 2 without a remainder)

Decision: first find the total number of outcomes. By condition, the subscriber remembers that one of the digits is zero, and the other digit is odd. Here it is more rational not to be wiser with combinatorics and use direct enumeration of outcomes . That is, when making a decision, we simply write down all the combinations:
01, 03, 05, 07, 09
10, 30, 50, 70, 90

And we count them - in total: 10 outcomes.

There is only one favorable outcome: the right number.

According to the classical definition:
is the probability that the subscriber will dial the correct number

Answer: 0,1

Decimal fractions in probability theory look quite appropriate, but you can also follow the traditional Vyshmatov style, operating only with ordinary fractions.

Advanced task for independent solution:

Task 4

The subscriber forgot the pin code for his SIM card, but remembers that it contains three "fives", and one of the numbers is either "seven" or "eight". What is the probability of successful authorization on the first attempt?

Here you can still develop the idea of ​​​​the probability that the subscriber is waiting for a punishment in the form of a puk-code, but, unfortunately, the reasoning will already go beyond the scope of this lesson.

Solution and answer below.

Sometimes listing combinations turns out to be a very painstaking task. In particular, this is the case in the next, no less popular group of problems, where 2 dice are thrown (less often - more):

Task 5

Find the probability that when two dice are thrown, the total will be:

a) five points
b) no more than four points;
c) from 3 to 9 points inclusive.

Decision: find the total number of outcomes:

Ways can drop the face of the 1st die and the face of the 2nd die can fall out in ways; on combination multiplication rule, Total: possible combinations. In other words, each the face of the 1st cube can be orderly couple with each face of the 2nd cube. We agree to write such a pair in the form , where is the number that fell on the 1st die, is the number that fell on the 2nd die. For example:

- 3 points on the first die, 5 points on the second, total points: 3 + 5 = 8;
- on the first die 6 points fell out, on the second - 1 point, the sum of points: 6 + 1 = 7;
- both dice rolled 2 points, sum: 2 + 2 = 4.

Obviously, the smallest amount is given by a pair, and the largest by two "sixes".

a) Consider the event: - when throwing two dice, 5 points will fall out. Let's write down and count the number of outcomes that favor this event:

Total: 4 favorable outcomes. According to the classical definition:
is the desired probability.

b) Consider the event: - no more than 4 points will fall out. That is, either 2, or 3, or 4 points. Again, we list and count the favorable combinations, on the left I will write down the total number of points, and after the colon - suitable pairs:

Total: 6 favorable combinations. Thus:
- the probability that no more than 4 points will fall out.

c) Let's consider the event: - from 3 to 9 points will fall out inclusive. Here you can go a straight road, but ... something does not feel like it. Yes, some pairs are already listed in the previous paragraphs, but there is still a lot of work to be done.

What's the best way to do it? In such cases, a detour turns out to be rational. Consider opposite event: - 2 or 10 or 11 or 12 points will fall out.

What's the point? The opposite event is favored by a much smaller number of pairs:

Total: 7 favorable outcomes.

According to the classical definition:
- the probability that less than three or more than 9 points will fall out.

In addition to direct enumeration and calculation of outcomes, various combinatorial formulas. And again the epic task about the elevator:

Task 7

3 people entered the elevator of a 20-storey building on the first floor. And let's go. Find the probability that:

a) they will go out on different floors
b) two will exit on the same floor;
c) everyone will exit on the same floor.

Our fascinating lesson has come to an end, and finally, once again, I strongly recommend, if not to solve, then at least to understand additional tasks on the classical definition of probability. As I noted, "stuffing the hand" also matters!

Further down the course - Geometric definition of probability and Theorems of addition and multiplication of probabilities and ... luck in the main!

Solutions and answers:

Task 2: Decision: 30 - 5 = 25 refrigerators have no defect.

is the probability that a randomly selected refrigerator does not have a defect.
Answer :

Task 4: Decision: find the total number of outcomes:
ways you can choose the place where the dubious figure is located and on each of these 4 places, 2 digits can be located (seven or eight). According to the rule of multiplication of combinations, the total number of outcomes: .
Alternatively, in the solution, you can simply list all the outcomes (fortunately there are not many of them):
7555, 8555, 5755, 5855, 5575, 5585, 5557, 5558
There is only one favorable outcome (correct pin code).
Thus, by the classical definition:
- the probability that the subscriber is authorized on the 1st attempt
Answer :

Task 6: Decision: find the total number of outcomes:
ways can drop numbers on 2 dice.

a) Consider the event: - when throwing two dice, the product of points will be equal to seven. For this event, there are no favorable outcomes, according to the classical definition of probability:
, i.e. this event is impossible.

b) Let's consider the event: - when throwing two dice, the product of points will be at least 20. This event is favored by the following outcomes:

Total: 8
According to the classical definition:
is the desired probability.

c) Consider opposite events:
– the product of points will be even;
– the product of points will be odd.
Let's list all the outcomes that favor the event:

Total: 9 favorable outcomes.
According to the classical definition of probability:
Opposite events form a complete group, so:
is the desired probability.

Answer :

Task 8: Decision: calculate the total number of outcomes: 10 coins can fall in ways.
Another way: 1st coin can fall in ways and 2nd coin can fall in ways and … and ways the 10th coin can fall. According to the rule of multiplying combinations, 10 coins can fall ways.
a) Consider the event: - all coins will fall heads. This event is favored by a single outcome, according to the classical definition of probability: .
b) Consider the event: - 9 coins will come up heads, and one will come up tails.
There are coins that can land tails. According to the classical definition of probability: .
c) Let's consider the following event: - heads will fall on half of the coins.
Exist unique combinations of five coins that can land heads. According to the classical definition of probability:
Answer :

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 may occur. The probability of an event A is denoted by P(A) (here P is the first letter of the French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favoring event A; - the number of all equally possible elementary outcomes of experience, forming a complete group of events.
This definition of probability is called classical. It arose at the initial stage of the development of probability theory.

The probability of an event has the following properties:
1. The probability of a certain event is equal to one. Let's designate a certain event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. We denote the impossible event by the letter . For an impossible event, therefore
(1.2.3)
3. The probability of a random event is expressed as a positive number less than one. Since the inequalities , or are satisfied for a random event, then
(1.2.4)
4. The probability of any event satisfies the inequalities
(1.2.5)
This follows from relations (1.2.2) -(1.2.4).

Example 1 An urn contains 10 balls of the same size and weight, of which 4 are red and 6 are blue. One ball is drawn from the urn. What is the probability that the drawn ball is blue?

Decision. The event "the drawn ball turned out to be blue" will be denoted by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor the event A. In accordance with formula (1.2.1), we obtain

Example 2 All natural numbers from 1 to 30 are written on identical cards and placed in an urn. After thoroughly mixing the cards, one card is removed from the urn. What is the probability that the number on the card drawn is a multiple of 5?

Decision. Denote by A the event "the number on the taken card is a multiple of 5". In this test, there are 30 equally possible elementary outcomes, of which 6 outcomes favor event A (numbers 5, 10, 15, 20, 25, 30). Hence,

Example 3 Two dice are thrown, the sum of points on the upper faces is calculated. Find the probability of the event B, consisting in the fact that the top faces of the cubes will have a total of 9 points.

Decision. There are 6 2 = 36 equally possible elementary outcomes in this trial. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), so

Example 4. A natural number not exceeding 10 is chosen at random. What is the probability that this number is prime?

Decision. Denote by the letter C the event "the chosen number is prime". In this case, n = 10, m = 4 (primes 2, 3, 5, 7). Therefore, the desired probability

Example 5 Two symmetrical coins are tossed. What is the probability that both coins have digits on the top sides?

Decision. Let's denote by the letter D the event "there was a number on the top side of each coin". There are 4 equally possible elementary outcomes in this test: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that on the first coin there is a coat of arms, on the second - a number). Event D is favored by one elementary outcome (C, C). Since m = 1, n = 4, then

Example 6 What is the probability that the digits in a randomly chosen two-digit number are the same?

Decision. Two-digit numbers are numbers from 10 to 99; there are 90 such numbers in total. 9 numbers have the same digits (these are the numbers 11, 22, 33, 44, 55, 66, 77, 88, 99). Since in this case m = 9, n = 90, then
,
where A is the "number with the same digits" event.

Example 7 From the letters of the word differential one letter is chosen at random. What is the probability that this letter will be: a) a vowel b) a consonant c) a letter h?

Decision. There are 12 letters in the word differential, of which 5 are vowels and 7 are consonants. Letters h this word does not. Let's denote the events: A - "vowel", B - "consonant", C - "letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n \u003d 12, then
, and .

Example 8 Two dice are tossed, the number of points on the top face of each dice is noted. Find the probability that both dice have the same number of points.

Decision. Let us denote this event by the letter A. Event A is favored by 6 elementary outcomes: (1;]), (2;2), (3;3), (4;4), (5;5), (6;6). In total there are equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. So the desired probability

Example 9 The book has 300 pages. What is the probability that a randomly opened page will have a sequence number that is a multiple of 5?

Decision. It follows from the conditions of the problem that there will be n = 300 of all equally possible elementary outcomes that form a complete group of events. Of these, m = 60 favor the occurrence of the specified event. Indeed, a number that is a multiple of 5 has the form 5k, where k is a natural number, and , whence . Hence,
, where A - the "page" event has a sequence number that is a multiple of 5".

Example 10. Two dice are thrown, the sum of points on the upper faces is calculated. What is more likely to get a total of 7 or 8?

Decision. Let's designate the events: A - "7 points fell out", B - "8 points fell out". Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B - by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). There are n = 6 2 = 36 of all equally possible elementary outcomes. Hence, and .

So, P(A)>P(B), that is, getting a total of 7 points is a more likely event than getting a total of 8 points.

Tasks

1. A natural number not exceeding 30 is chosen at random. What is the probability that this number is a multiple of 3?
2. In the urn a red and b blue balls of the same size and weight. What is the probability that a randomly drawn ball from this urn is blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of zo?
4. In the urn a blue and b red balls of the same size and weight. One ball is drawn from this urn and set aside. This ball is red. Then another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A natural number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are thrown, the sum of points on the upper faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed, the sum of the dropped points is calculated. What is more likely to get a total of 11 (event A) or 12 points (event B)?

Answers

1. 1/3. 2 . b/(a+b). 3 . 0,2. 4 . (b-1)/(a+b-1). 5 .0,3.6 . p 1 \u003d 25/216 - the probability of getting 9 points in total; p 2 \u003d 27/216 - the probability of getting 10 points in total; p2 > p1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).

Questions

1. What is called the probability of an event?
2. What is the probability of a certain event?
3. What is the probability of an impossible event?
4. What are the limits of the probability of a random event?
5. What are the limits of the probability of any event?
6. What definition of probability is called classical?

Fundamentals of Probability Theory

Plan:

1. Random events

2. Classical definition of probability

3. Calculation of event probabilities and combinatorics

4. Geometric probability

Theoretical information

Random events.

random phenomenon- a phenomenon, the outcome of which is unambiguously determined. This concept can be interpreted in a fairly broad sense. Namely: everything in nature is quite accidental, the appearance and birth of any individual is a random phenomenon, the choice of goods in a store is also a random phenomenon, getting a mark 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. Hitting it on the target is accidental, but hitting a projectile in a certain "fork" is a pattern. You can specify the distance closer than and beyond which the projectile will not fly. Get some "fork dispersion of shells"

~ The same body is weighed several times. Strictly speaking, different results will be obtained each time, albeit differing by a negligibly small amount, but different.

~ An aircraft flying along the same route has a certain flight corridor within which the aircraft can maneuver, but it will never have exactly the same route

~ An athlete will never be able to run the same distance with the same time. His results will also be within a certain numerical range.

Experience, experiment, observation are tests

Trial- observation or fulfillment of a certain set of conditions that are performed repeatedly, and regularly repeated in this and the same sequence, duration, while observing other identical parameters.

Let's consider performance by the sportsman of a shot on 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 "miss" are events as a result of a shot.

Event– qualitative test result.

An event may or may not occur Events are indicated by capital Latin letters. For example: D ="The shooter hit the target". S="White ball drawn". K="Random lottery ticket without winning.".

Tossing a coin is a test. 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 needed at a given time by the researcher, while others may not be needed.

The event is called random, if under the implementation of a certain set of conditions S it can either happen or not happen. In what follows, instead of saying "the set of conditions S is fulfilled," we will say briefly: "the test was carried out." Thus, the event will be considered as the result of the test.

~ The shooter shoots at a target divided into four areas. The shot is a test. Hitting a certain area of ​​the target is an event.

~ There are colored balls in the urn. One ball is drawn at random from the urn. Removing a ball from an urn is a test. The appearance of a ball of a certain color is an event.

Types of random events

1. Events are said to be incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.

~ A part was taken at random from a box with parts. The appearance of a standard part excludes the appearance of a non-standard part. Events € a standard part appeared" and with a non-standard part appeared" - incompatible.

~ A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events "a coat of arms appeared" and "an inscription appeared" are incompatible.

Several events form full group, if at least one of them appears as a result of the test. In other words, the occurrence of at least one of the events of the complete group is a certain event.

In particular, if the events that form a complete group are pairwise incompatible, 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 tickets of the money and clothing lottery were purchased. One and only one of the following events must occur:

1. "the winnings fell on the first ticket and did not fall on the second",

2. "the winnings did not fall on the first ticket and fell on the second",

3. "the winnings fell on both tickets",

4. "both tickets did not win."

These events form a complete group of pairwise incompatible events,

~ The shooter fired a shot at the target. One of the following two events is sure to occur: hit, miss. These two disjoint events also form a complete group.

2. Events are called equally possible if there is reason to believe that neither is more possible than the other.

~ The appearance of a "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 a coinage does not affect the loss of one or another side of the coin.

~ The appearance of one or another number of points on a thrown dice is an equally probable event. Indeed, it is assumed that the die is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of points does not affect the loss of any face.

3. The event is called authentic, if it cannot 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 the given event. Opposite events are not compatible, but one of them must necessarily occur. Opposite events are commonly referred to as negations, i.e. a dash is written above the letter. The events are opposite: A and Ā; U and Ū, etc. .

The classical definition of probability

Probability is one of the basic concepts of probability theory.

There are several definitions of this concept. Let us give a definition that is called classical. Next, we point out the weaknesses of this definition and give other definitions that make it possible to overcome the shortcomings of the classical definition.

Consider the situation: A box contains 6 identical balls, 2 being red, 3 being blue and 1 being white. Obviously, the possibility of drawing a colored (i.e., red or blue) ball at random from an urn is greater than the possibility of drawing a white ball. This possibility can be characterized by a number, which is called the probability of an event (the appearance of a colored ball).

Probability- a number characterizing the degree of possibility of occurrence of the event.

In the situation under consideration, we denote:

Event A = "Pulling out a colored ball".

Each of the possible outcomes of the test (the test consists in extracting a ball from the urn) is called elementary (possible) outcome and event. Elementary outcomes can be denoted by letters with indexes below, for example: k 1 , k 2 .

In our example, there are 6 balls, so there are 6 possible outcomes: a white ball appeared; a red ball appeared; a blue ball appeared, and so on. It is easy to see that these outcomes form a complete group of pairwise incompatible events (only one ball will necessarily appear) and they are equally probable (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, we will call favorable outcomes this event. In our example, the event is favored BUT(the appearance of a colored ball) the following 5 outcomes:

Thus the event BUT observed if one occurs in the test, no matter which, of the elementary outcomes that favor BUT. This is the appearance of any colored ball, of which there are 5 pieces in the box

In the considered example of elementary outcomes 6; of which 5 favor the event BUT. Hence, P(A)= 5/6. This number gives that quantification of the degree of possibility of the appearance of a colored ball.

Probability definition:

Probability of event A is the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form a complete group.

P(A)=m/n or P(A)=m: n, where:

m is the number of elementary outcomes that favor BUT;

P- the number of all possible elementary outcomes of the test.

It is assumed here that the elementary outcomes are incompatible, equally probable 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 each elementary outcome of the test favors the event. In this case m = n hence 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 trial favors the event. In this case m=0, hence p=0.

3.The probability of a random event is a positive number between zero and one. 0t< n.

In subsequent topics, theorems will be given that allow us to find the probabilities of other events from the known probabilities of some events.

Measurement. There are 6 girls and 4 boys in the group of students. What is the probability that a randomly selected student will be a girl? will it be a young man?

p dev = 6 / 10 = 0.6 p jun = 4 / 10 = 0.4

The concept of "probability" in modern rigorous courses of probability theory is built on a set-theoretic basis. Let's take a look at some of this approach.

Suppose that as a result of the test one and only one of the following events occurs: w i(i=1, 2, .... n). Events w i, is called elementary events (elementary outcomes). O it follows that the elementary events are pairwise incompatible. The set of all elementary events that can appear in a trial is called elementary event spaceΩ (Greek letter omega capital), and the elementary events themselves - points in this space..

Event BUT identified with a subset (of the space Ω) whose elements are elementary outcomes favoring BUT; event AT is a subset Ω whose elements are outcomes that favor AT, etc. Thus, the set of all events that can occur in the test is the set of all subsets of Ω. Ω itself occurs for any outcome of the test, therefore Ω is a certain 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 distinguished from among all events by topics, "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 equal to 1 or with the sign of the sum, this fact will be written as an expression:

By definition, the probability P(A) events BUT is equal to the sum of the probabilities of elementary outcomes favoring BUT. Therefore, the probability of a certain event is equal to one, impossible - to zero, arbitrary - is between zero and one.

Let us consider an important particular case, when all outcomes are equally probable. The number of outcomes is equal to l, the sum of the probabilities of all outcomes is equal to one; hence the probability of each outcome is 1/n. Let the event BUT favors m outcomes.

Event Probability BUT is equal to the sum of the probabilities of outcomes favoring BUT:

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., undefined concepts are elementary event and probability. The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its probability.

Here are the axioms that define the probability:

1. Every event BUT assigned a non-negative real number P(A). This number is called the probability of the event. BUT.

2. The probability of a certain 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 for the relationship between them are derived as theorems.

MUNICIPAL EDUCATIONAL INSTITUTION

GYMNASIUM No. 6

on the topic "Classical definition of probability".

Completed by a student of the 8th "B" class

Klimantova Alexandra.

Mathematics teacher: Videnkina V. A.

Voronezh, 2008

Many games use a dice. The die has 6 faces, on each face a different number of points is marked - from 1 to 6. The player throws the die and looks at how many points there are on the dropped face (on the face that is located on top). Quite often, the dots on the edge of the die are replaced by the corresponding number and then they talk about a roll of 1, 2 or 6. Throwing a die can be considered an experience, an experiment, a test, and the result obtained is the outcome of a test or an elementary event. People are interested in guessing the onset of an event, predicting its outcome. What predictions can they make when a dice is rolled? For example, these:

1. event A - the number 1, 2, 3, 4, 5 or 6 falls out;
2. event B - the number 7, 8 or 9 falls out;
3. event C - the number 1 falls out.

Event A, predicted in the first case, will definitely come. In general, an event that is sure to occur in a given experience is called certain event.

Event B, predicted in the second case, will never occur, it is simply impossible. In general, an event that cannot occur in a given experiment is called impossible event.

Will the event C, predicted in the third case, happen or not? We are not able to answer this question with complete certainty, since 1 may or may not fall out. An event that in a given experience may or may not occur is called random event.

Thinking about the onset of a certain event, we most likely will not use the word “probably”. For example, if today is Wednesday, then tomorrow is Thursday, this is a certain event. On Wednesday we will not say: "Probably tomorrow is Thursday", we will say briefly and clearly: "Tomorrow is Thursday." True, if we are prone to beautiful phrases, then we can say this: "With one hundred percent probability I say that tomorrow is Thursday." On the contrary, if today is Wednesday, then the coming of tomorrow is Friday—an impossible event. Evaluating this event on Wednesday, we can say this: "I'm sure that tomorrow is not Friday." Or like this: "It's unbelievable that tomorrow is Friday." Well, if we are prone to beautiful phrases, then we can say this: “The probability that tomorrow is Friday is zero.” So, a certain event is an event that occurs under given conditions. with 100% certainty(i.e. coming in 10 cases out of 10, in 100 cases out of 100, etc.). An impossible event is an event that never occurs under given conditions, an event with zero probability.

But, unfortunately (and perhaps fortunately), not everything in life is so clear and clear: it will always be (certain event), this will never happen (impossible event). Most often, we are faced with random events, some of which are more likely, others less likely. Usually people use the words "more likely" or "less likely", as they say, on a whim, relying on what is called common sense. But very often such estimates turn out to be insufficient, since it is important to know how much percent likely a random event or how many times one random event is more likely than another. In other words, we need exact quantitative characteristics, you need to be able to characterize the probability by a number.

We have already taken the first steps in this direction. We said that the probability of a certain event occurring is characterized as one hundred percent, and the probability of an impossible event occurring as zero. Given that 100% equals 1, people have agreed on the following:

1. the probability of a certain event is considered to be equal to 1;
2. the probability of an impossible event is considered equal to 0.

How do you calculate the probability of a random event? After all, it happened by chance, which means that it does not obey laws, algorithms, formulas. It turns out that certain laws operate in the world of randomness, allowing you to calculate probabilities. This is the branch of mathematics that is called- probability theory.

Mathematics deals with model some phenomenon of the reality around us. Of all the models used in probability theory, we will limit ourselves to the simplest.

Classical probabilistic scheme

To find the probability of an event A during some experiment, one should:

1) find the number N of all possible outcomes of this experience;

2) accept the assumption that all these outcomes are equally probable (equally possible);

3) find the number N(A) of those outcomes of the experience in which the event A occurs;

4) find a private ; it will be equal to the probability of event A.

It is customary to designate the probability of an event A as P(A). The explanation for this designation is very simple: the word "probability" in French is probability, in English- probability.The designation uses the first letter of the word.

Using this notation, the probability of an event A according to the classical scheme can be found using the formula

P(A)=.

Often all the points of the given classical probabilistic scheme are expressed in one rather long phrase.

The classical definition of probability

The probability of an event A during a certain test is the ratio of the number of outcomes, as a result of which the event A occurs, to the total number of all equally possible outcomes of this test.

Example 1. Find the probability that in one throw of a dice: a) 4; b) 5; c) an even number of points; d) the number of points greater than 4; e) number of points not a multiple of three.

Decision. In total, there are N=6 possible outcomes: dropping a face of a cube with a number of points equal to 1, 2, 3, 4, 5, or 6. We believe that none of them has any advantages over the others, i.e., we accept the assumption of the similarity of these outcomes.

a) Exactly in one of the outcomes, the event of interest to us A will occur - the loss of the number 4. Hence, N (A) \u003d 1 and

P(A)= =.

b) The solution and the answer are the same as in the previous paragraph.

c) The event B of interest to us will occur exactly in three cases when the number of points is 2, 4 or 6. Hence,

N(B)=3 andP(B)==.

d) The event C of interest to us will occur exactly in two cases when the number of points is 5 or 6. Hence,

N(C) =2 and P(C)=.

e) Of the six possible numbers drawn, four (1, 2, 4 and 5) are not multiples of three, and the remaining two (3 and 6) are divisible by three. This means that the event of interest to us occurs exactly in four out of six possible and equally probable among themselves and equally probable among themselves outcomes of the experience. So the answer is .

Answer: a); b) ; in) ; G) ; e).

A real playing dice may well differ from an ideal (model) dice, therefore, to describe its behavior, a more accurate and detailed model is required, taking into account the advantages of one face over another, the possible presence of magnets, etc. But “the devil is in the details”, and more accuracy tends to lead to more complexity, and getting an answer becomes a problem. We confine ourselves to considering the simplest probabilistic model, where all possible outcomes are equally probable.

Remark 1. Let's consider another example. The question was asked: "What is the probability of getting a three on one roll of the die?" The student answered like this: "The probability is 0.5." And he explained his answer: “The three will either fall out or not. This means that there are two outcomes in total, and in exactly one event the event of interest to us occurs. According to the classical probabilistic scheme, we get the answer 0.5. Is there an error in this reasoning? At first glance, no. However, it is still there, and in a fundamental moment. Yes, indeed, the triple will either fall out or not, that is, with such a definition of the outcome of the throw, N = 2. It is also true that N(A)=1 and, of course, it is true that =0, 5, i.e., three points of the probabilistic scheme are taken into account, but the fulfillment of point 2) is doubtful. Of course, from a purely legal point of view, we have the right to believe that the loss of a triple is equally likely to fail. But can we think so without violating our own natural assumptions about the "sameness" of the faces? Of course not! Here we are dealing with correct reasoning within some model. Only this model itself is “wrong”, not corresponding to the real phenomenon.

Remark 2. When discussing probability, do not lose sight of the following important circumstance. If we say that when throwing a die, the probability of getting one point is equal to , this does not mean at all that by rolling the die 6 times, you will get one point exactly once, by throwing the die 12 times, you will get one point exactly twice, by rolling the die 18 times, you get one point exactly three times, and so on. The word is probably speculative. We assume that is likely to happen. Probably if we roll the die 600 times, one point will come up 100 times, or about 100.

Probability theory arose in the 17th century when analyzing various gambling games. It is not surprising, therefore, that the first examples are of a playful nature. From the dice examples, let's move on to the random drawing of playing cards from the deck.

Example 2. From a deck of 36 cards, 3 cards are randomly drawn at the same time. What is the probability that there is no Queen of Spades among them?

Decision. We have a set of 36 elements. We select three elements, the order of which is not important. Hence, it is possible to obtain N=C outcomes. We will act according to the classical probabilistic scheme, that is, we will assume that all these outcomes are equally probable.

It remains to calculate the required probability according to the classical definition:

And what is the probability that among the chosen three cards there is a Queen of Spades? The number of all such outcomes is not difficult to calculate, you just need to subtract from all outcomes N all those outcomes in which there is no queen of spades, that is, subtract the number N(A) found in Example 3. Then this difference N - N (A) in accordance with the classical probabilistic scheme should be divided by N. This is what we get:

We see that there is a certain relationship between the probabilities of the two events. If event A consists in the absence of the Queen of Spades, and event B consists in her presence among the chosen three cards, then

P (B) \u003d 1 - P (A),

P(A)+P(B)=1.

Unfortunately, in the equality P(A)+P(B)=1 there is no information about the relationship between events A and B; we have to keep this connection in mind. It would be more convenient to give the event B a name and designation in advance, clearly indicating its connection with A.

Definition 1. Event B called opposite to event A and denote B=Ā if event B occurs if and only if event A does not occur.

TTheorem 1. To find the probability of the opposite event, subtract the probability of the event itself from unity: Р(Ā)= 1—Р(А). Indeed,

In practice, they calculate what is easier to find: either P(A) or P(Ā). After that, they use the formula from the theorem and find, respectively, either P(Ā)= 1-P(A), or P(A)= 1-P(Ā).

Often used is the method of solving a particular problem by "enumeration of cases", when the conditions of the problem are divided into mutually exclusive cases, each of which is considered separately. For example, “if you go to the right, you will lose your horse, if you go straight, you will solve a problem according to probability theory, if you go to the left…”. Or when plotting the function y=│x+1│—│2x—5│, consider the cases of x

Example 3. Of the 50 dots, 17 are shaded blue and 13 are orange. Find the probability that a randomly selected point will be shaded.

Decision. In total, 30 points out of 50 are shaded. Hence, the probability is = 0.6.

Answer: 0.6.

Let's take a closer look at this simple example, however. Let event A be that the selected point is blue, and event B be that the selected point is orange. By convention, events A and B cannot happen at the same time.

We denote by the letter C the event of interest to us. Event C occurs if and only if it occurs at least one of the events A or B. It is clear that N(C)= N(A)+N(B).

Let us divide both sides of this equality by N, the number of all possible outcomes of the given experiment; we get

We have analyzed an important and frequently occurring situation using a simple example. There is a special name for her.

Definition 2. Events A and B are called incompatible if they cannot occur at the same time.

Theorem 2. The probability of occurrence of at least one of two incompatible events is equal to the sum of their probabilities.

When translating this theorem into mathematical language, it becomes necessary to somehow name and designate an event consisting in the occurrence of at least one of the two given events A and B. Such an event is called the sum of events A and B and denoted by A+B.

If A and B are incompatible, then P(A+B)= P(A)+P(B).

Indeed,

The incompatibility of events A and B can be conveniently illustrated by a figure. If all the outcomes of the experience are some set of points in the figure, then the events A and B are some subsets of a given set. The incompatibility of A and B means that these two subsets do not intersect. A typical example of incompatible events is any event A and the opposite event Ā.

Of course, this theorem is true for three, four, and for any finite number of pairwise incompatible events. The probability of the sum of any number of pairwise incompatible events is equal to the sum of the probabilities of these events. This important statement exactly corresponds to the method of solving problems by "enumeration of cases".

Between the events that occur as a result of some experience, and between the probabilities of these events, there may be some relationships, dependencies, connections, etc. For example, events can be “added”, and the probability of the sum of incompatible events is equal to the sum of their probabilities.

In conclusion, we discuss the following fundamental question: is it possible to prove, that the probability of getting "tails" in one toss of a coin is equal to

The answer is negative. Generally speaking, the question itself is not correct, the exact meaning of the word "prove" is not clear. After all, we always prove something within the framework of some models, in which the rules, laws, axioms, formulas, theorems, etc. are already known. If we are talking about an imaginary, “ideal” coin, then that is why it is considered ideal because, a-priory, the probability of getting heads is equal to the probability of getting heads. And, in principle, we can consider a model in which the probability of falling “tails” is twice the probability of falling “heads”, or three times less, etc. Then the question arises: for what reason from the various possible models for tossing a coin do we choose one in which both outcomes of the toss are equally likely?

A completely frontal answer is: “But it’s easier, clearer and more natural for us!” But there are more substantive arguments as well. They come from practice. The vast majority of textbooks on probability theory give examples of the French naturalist J. Buffon (18th century) and the English mathematician-statistician C. Pearson (late 19th century), who threw a coin 4040 and 24000 times, respectively, and counted the number of falling “eagles ” or “tails”. Their “tails” fell out, respectively, 1992 and 11998 times. If you count drop frequency“tails”, then you get = = 0.493069 ... for Buffon and = 0.4995 for Pearson. Arises naturally assumption that with an unlimited increase in the number of tosses of a coin, the frequency of falling "tails", as well as the frequency of falling "eagles", will more and more approach 0.5. It is this assumption, based on practical data, that is the basis for choosing a model with equiprobable outcomes.

Now we can sum up. The basic concept is probability of a random event, which is calculated within the framework of the simplest model— classical probabilistic scheme. The concept is important both in theory and in practice. opposite event and the formula Р(Ā)= 1—Р(А) for finding the probability of such an event.

Finally, we met incompatible events and with formulas.

P (A + B) \u003d P (A) + P (B),

P (A + B + C) \u003d P (A) + P (B) + P (C),

allowing to find probabilities amounts such events.

Bibliography

1. Events. Probabilities. Statistical data processing: Add. paragraphs to the course of algebra 7-9 cells. educational institutions / A. G. Mordkovich, P. V. Semenov.—4th ed.—M.: Mnemozina, 2006.—112 p.: ill.

2.Yu. N. Makarychev, N. G. Mindyuk “Algebra. Elements of statistics and probability theory.—Moscow, Enlightenment, 2006.