In what case do the outcomes occur? Probability theory: formulas and examples of problem solving. Classical probabilistic scheme
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 an experimental characteristic.
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 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 favorable for a given event to the number of all equally possible outcomes of the experience in which this event may appear.
The probability of event A is denoted by P(A)(Here R– the first letter of a French word probabilite- probability).
According to the definition
where is the number of elementary test outcomes favorable to the occurrence of the event;
The total number of possible elementary test outcomes.
This definition of probability is called classic. It arose at the initial stage of the development of probability theory.
The number is often called the relative frequency of occurrence of an event A in experience.
The greater the probability of an event, the more often it occurs, and vice versa, the less probability of an event, the less often it occurs. When the probability of an event is close to 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 occurrence.
On the contrary, 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: P(A) 100% is the average percentage of the number of occurrences of an event A.
Example 2.13. While dialing a phone number, the subscriber forgot one digit and dialed it at random. Find the probability that the correct number is dialed.
Solution.
Let us denote by A event - “the required number has been 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 A only one outcome (there is only one required number).
The required probability is equal to the ratio of the number of outcomes favorable to the event to the number of all elementary outcomes:
The classical probability formula provides a very simple, experiment-free way to calculate probabilities. However, the simplicity of this formula is very deceptive. The fact is that when using it, two very difficult questions usually arise:
1. How to choose a system of experimental outcomes so that they are equally possible, and is it possible to do this at all?
2. How to find numbers m And n?
If several objects are involved in an experiment, it is not always easy to see equally possible 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 equipossibility of outcomes in an experiment with only two coins!
Example 2.14. ( d'Alembert's error). Two identical coins are tossed. What is the probability that they will fall on the same side?
D'Alembert's solution.
The experiment has three equally possible outcomes:
1. Both coins will land on heads;
2. Both coins will land on tails;
3. One of the coins will land on heads, the other on tails.
Correct solution.
The experiment has four equally possible outcomes:
1. The first coin will fall on heads, the second will also fall on heads;
2. The first coin will land on tails, the second will also land on tails;
3. The first coin will fall 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 required 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.
“Accidents are not accidental”... It sounds like something a philosopher said, but in fact, studying randomness is the destiny of the great science of mathematics. In mathematics, chance is dealt with by probability theory. Formulas and examples of tasks, as well as the main definitions of this science will be presented in the article.
What is probability theory?
Probability theory is one of the mathematical disciplines that studies random events.
To make it a little clearer, let's give a small example: if you throw a coin up, it can land on heads or tails. While the coin is in the air, both of these probabilities are possible. That is, the probability of possible consequences is 1:1. If one is drawn from a deck of 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict here, especially with the help of mathematical formulas. However, if you repeat a certain action many times, you can identify a certain pattern and, based on it, predict the outcome of events in other conditions.
To summarize all of the above, probability theory in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical value.
From the pages of history
The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.
Initially, probability theory had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. They studied gambling for a long time and saw certain patterns, which they decided to tell the public about.
The same technique was invented by Christiaan Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of “probability theory”, formulas and examples, which are considered the first in the history of the discipline, were introduced by him.
The works of Jacob Bernoulli, Laplace's and Poisson's theorems are also of no small importance. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks received their current form thanks to Kolmogorov’s axioms. As a result of all the changes, probability theory became one of the mathematical branches.
Basic concepts of probability theory. Events
The main concept of this discipline is “event”. There are three types of events:
- Reliable. Those that will happen anyway (the coin will fall).
- Impossible. Events that will not happen under any circumstances (the coin will remain hanging in the air).
- Random. The ones that will happen or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then there are random factors that can affect the result: the physical characteristics of the coin, its shape, its original position, the force of the throw, etc.
All events in the examples are indicated in capital Latin letters, with the exception of P, which has a different role. For example:
- A = “students came to lecture.”
- Ā = “students did not come to the lecture.”
In practical tasks, events are usually written down in words.
One of the most important characteristics of events is their equal possibility. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally possible. This happens when someone deliberately influences an outcome. For example, “marked” playing cards or dice, in which the center of gravity is shifted.
Events can also be compatible and incompatible. Compatible events do not exclude each other's occurrence. For example:
- A = “the student came to the lecture.”
- B = “the student came to the lecture.”
These events are independent of each other, and the occurrence of one of them does not affect the occurrence of the other. Incompatible events are defined by the fact that the occurrence of one excludes the occurrence of another. If we talk about the same coin, then the loss of “tails” makes it impossible for the appearance of “heads” in the same experiment.
Actions on events
Events can be multiplied and added; accordingly, logical connectives “AND” and “OR” are introduced in the discipline.
The amount is determined by the fact that either event A or B, or two, can occur simultaneously. If they are incompatible, the last option is impossible; either A or B will be rolled.
Multiplication of events consists in the appearance of A and B at the same time.
Now we can give several examples to better remember the basics, probability theory and formulas. Examples of problem solving below.
Exercise 1: The company takes part in a competition to receive contracts for three types of work. Possible events that may occur:
- A = “the firm will receive the first contract.”
- A 1 = “the firm will not receive the first contract.”
- B = “the firm will receive a second contract.”
- B 1 = “the firm will not receive a second contract”
- C = “the firm will receive a third contract.”
- C 1 = “the firm will not receive a third contract.”
Using actions on events, we will try to express the following situations:
- K = “the company will receive all contracts.”
In mathematical form, the equation will have the following form: K = ABC.
- M = “the company will not receive a single contract.”
M = A 1 B 1 C 1.
Let’s complicate the task: H = “the company will receive one contract.” Since it is not known which contract the company will receive (first, second or third), it is necessary to record the entire range of possible events:
H = A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.
And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second. Other possible events were recorded using the appropriate method. The symbol υ in the discipline denotes the connective “OR”. If we translate the above example into human language, the company will receive either the third contract, or the second, or the first. In a similar way, you can write down other conditions in the discipline “Probability Theory”. The formulas and examples of problem solving presented above will help you do this yourself.
Actually, the probability
Perhaps, in this mathematical discipline, the probability of an event is the central concept. There are 3 definitions of probability:
- classic;
- statistical;
- geometric.
Each has its place in the study of probability. Probability theory, formulas and examples (9th grade) mainly use the classical definition, which sounds like this:
- The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.
The formula looks like this: P(A)=m/n.
A is actually an event. If a case opposite to A appears, it can be written as Ā or A 1 .
m is the number of possible favorable cases.
n - all events that can happen.
For example, A = “draw a card of the heart suit.” There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:
P(A)=9/36=0.25.
As a result, the probability that a card of the heart suit will be drawn from the deck will be 0.25.
Towards higher mathematics
Now it has become a little known what the theory of probability is, formulas and examples of solving problems that come across in the school curriculum. However, probability theory is also found in higher mathematics, which is taught in universities. Most often they operate with geometric and statistical definitions of the theory and complex formulas.
The theory of probability is very interesting. It is better to start studying formulas and examples (higher mathematics) small - with the statistical (or frequency) definition of probability.
The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic one:
If the classical formula is calculated for prediction, then the statistical one is calculated according to the results of the experiment. Let's take a small task for example.
The technological control department checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?
A = “the appearance of a quality product.”
W n (A)=97/100=0.97
Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Out of 100 products that were checked, 3 were found to be of poor quality. We subtract 3 from 100 and get 97, this is the amount of quality goods.
A little about combinatorics
Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B can be made in n different ways, then the choice of A and B can be made by multiplication.
For example, there are 5 roads leading from city A to city B. There are 4 paths from city B to city C. In how many ways can you get from city A to city C?
It's simple: 5x4=20, that is, in twenty different ways you can get from point A to point C.
Let's complicate the task. How many ways are there to lay out cards in solitaire? There are 36 cards in the deck - this is the starting point. To find out the number of ways, you need to “subtract” one card at a time from the starting point and multiply.
That is, 36x35x34x33x32...x2x1= the result does not fit on the calculator screen, so it can simply be designated 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied together.
In combinatorics there are such concepts as permutation, placement and combination. Each of them has its own formula.
An ordered set of elements of a set is called an arrangement. Placements can be repeated, that is, one element can be used several times. And without repetition, when elements are not repeated. n are all elements, m are elements that participate in the placement. The formula for placement without repetition will look like:
A n m =n!/(n-m)!
Connections of n elements that differ only in the order of placement are called permutations. In mathematics it looks like: P n = n!
Combinations of n elements of m are those compounds in which it is important what elements they were and what their total number is. The formula will look like:
A n m =n!/m!(n-m)!
Bernoulli's formula
In probability theory, as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the occurrence of A in an experiment does not depend on the occurrence or non-occurrence of the same event in earlier or subsequent trials.
Bernoulli's equation:
P n (m) = C n m ×p m ×q n-m.
The probability (p) of the occurrence of event (A) is constant for each trial. The probability that the situation will occur exactly m times in n number of experiments will be calculated by the formula presented above. Accordingly, the question arises of how to find out the number q.
If event A occurs p number of times, accordingly, it may not occur. Unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that denotes the possibility of an event not occurring.
Now you know Bernoulli's formula (probability theory). We will consider examples of problem solving (first level) below.
Task 2: A store visitor will make a purchase with probability 0.2. 6 visitors independently entered the store. What is the likelihood that a visitor will make a purchase?
Solution: Since it is unknown how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.
A = “the visitor will make a purchase.”
In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.
n = 6 (since there are 6 customers in the store). The number m will vary from 0 (not a single customer will make a purchase) to 6 (all visitors to the store will purchase something). As a result, we get the solution:
P 6 (0) = C 0 6 ×p 0 ×q 6 =q 6 = (0.8) 6 = 0.2621.
None of the buyers will make a purchase with probability 0.2621.
How else is Bernoulli's formula (probability theory) used? Examples of problem solving (second level) below.
After the above example, questions arise about where C and r went. Relative to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:
C n m = n! /m!(n-m)!
Since in the first example m = 0, respectively, C = 1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of two visitors purchasing goods.
P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.
The theory of probability is not that complicated. Bernoulli's formula, examples of which are presented above, is direct proof of this.
Poisson's formula
Poisson's equation is used to calculate low probability random situations.
Basic formula:
P n (m)=λ m /m! × e (-λ) .
In this case λ = n x p. Here is a simple Poisson formula (probability theory). We will consider examples of problem solving below.
Task 3: The factory produced 100,000 parts. Occurrence of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?
As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of solving problems of this kind are no different from other tasks in the discipline; we substitute the necessary data into the given formula:
A = “a randomly selected part will be defective.”
p = 0.0001 (according to the task conditions).
n = 100000 (number of parts).
m = 5 (defective parts). We substitute the data into the formula and get:
R 100000 (5) = 10 5 /5! X e -10 = 0.0375.
Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In fact, it can be found by the formula:
e -λ = lim n ->∞ (1-λ/n) n .
However, there are special tables that contain almost all values of e.
De Moivre-Laplace theorem
If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of occurrence of event A in all schemes is the same, then the probability of occurrence of event A a certain number of times in a series of tests can be found by Laplace’s formula:
Р n (m)= 1/√npq x ϕ(X m).
X m = m-np/√npq.
To better remember Laplace’s formula (probability theory), examples of problems are below to help.
First, let's find X m, substitute the data (they are all listed above) into the formula and get 0.025. Using tables, we find the number ϕ(0.025), the value of which is 0.3988. Now you can substitute all the data into the formula:
P 800 (267) = 1/√(800 x 1/3 x 2/3) x 0.3988 = 3/40 x 0.3988 = 0.03.
Thus, the probability that the flyer will work exactly 267 times is 0.03.
Bayes formula
The Bayes formula (probability theory), examples of solving problems with the help of which will be given below, is an equation that describes the probability of an event based on the circumstances that could be associated with it. The basic formula is as follows:
P (A|B) = P (B|A) x P (A) / P (B).
A and B are definite events.
P(A|B) is a conditional probability, that is, event A can occur provided that event B is true.
P (B|A) - conditional probability of event B.
So, the final part of the short course “Probability Theory” is the Bayes formula, examples of solutions to problems with which are below.
Task 5: Phones from three companies were brought to the warehouse. At the same time, the share of phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. You need to find the probability that a randomly selected phone will be defective.
A = “randomly picked phone.”
B 1 - the phone that the first factory produced. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).
As a result we get:
P (B 1) = 25%/100% = 0.25; P(B 2) = 0.6; P (B 3) = 0.15 - thus we found the probability of each option.
Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in companies:
P (A/B 1) = 2%/100% = 0.02;
P(A/B 2) = 0.04;
P (A/B 3) = 0.01.
Now let’s substitute the data into the Bayes formula and get:
P (A) = 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 = 0.0305.
The article presents probability theory, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after everything that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It’s difficult for an ordinary person to answer; it’s better to ask someone who has used it to win the jackpot more than once.
In economics, as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the sales volume of a product depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and carrying out sales, you have to predict the outcome of such activities on the basis of either your own previous experience, or similar experience of other people, or intuition, which to a large extent also relies on experimental data.
In order to somehow evaluate the event in question, it is necessary to take into account or specially organize the conditions in which this event is recorded.
The implementation of certain conditions or actions to identify the event in question is called experience or experiment.
The event is called random, if as a result of experience it may or may not occur.
The event is called reliable, if it necessarily appears as a result of a given experience, and impossible, if it cannot appear in this experience.
For example, snowfall in Moscow on November 30 is a random event. The daily sunrise can be considered a reliable event. Snowfall at the equator can be considered an impossible event.
One of the main tasks in probability theory is the task of determining a quantitative measure of the possibility of an event occurring.
Algebra of events
Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.
Amount events is an event consisting of the occurrence of at least one of these events
An example of the sum of events is the presence of at least one of two products in the store.
The work events is an event consisting of the simultaneous occurrence of all these events
An event consisting of the appearance of two goods in a store at the same time is a product of events: - the appearance of one product, - the appearance of another product.
Events form a complete group of events if at least one of them is sure to occur in experience.
Example. The port has two berths for receiving ships. Three events can be considered: - the absence of ships at the berths, - the presence of one ship at one of the berths, - the presence of two ships at two berths. These three events form a complete group of events.
Opposite two unique possible events that form a complete group are called.
If one of the events that is opposite is denoted by , then the opposite event is usually denoted by .
Classical and statistical definitions of event probability
Each of the equally possible results of tests (experiments) is called an elementary outcome. They are usually designated by letters. For example, a die is thrown. There can be a total of six elementary outcomes based on the number of points on the sides.
From elementary outcomes you can create a more complex event. Thus, the event of an even number of points is determined by three outcomes: 2, 4, 6.
A quantitative measure of the possibility of the occurrence of the event in question is probability.
The most widely used definitions of the probability of an event are: classic And statistical.
The classical definition of probability is associated with the concept of a favorable outcome.
The outcome is called favorable to a given event if its occurrence entails the occurrence of this event.
In the above example, the event in question—an even number of points on the rolled side—has three favorable outcomes. In this case, the general
number of possible outcomes. This means that the classical definition of the probability of an event can be used here.
Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes
where is the probability of the event, is the number of outcomes favorable to the event, is the total number of possible outcomes.
In the considered example
The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.
The relative frequency of occurrence of an event is calculated using the formula
where is the number of occurrences of an event in a series of experiments (tests).
Statistical definition. The probability of an event is the number around which the relative frequency stabilizes (sets) with an unlimited increase in the number of experiments.
In practical problems, the probability of an event is taken to be the relative frequency for a sufficiently large number of trials.
From these definitions of the probability of an event it is clear that the inequality is always satisfied
To determine the probability of an event based on formula (1.1), combinatorics formulas are often used, which are used to find the number of favorable outcomes and the total number of possible outcomes.
MUNICIPAL EDUCATIONAL INSTITUTION
GYMNASIUM No. 6
on the topic “Classical definition of probability.”
Completed by a student of grade 8 "B"
Klimantova Alexandra.
Mathematics teacher: Videnkina V. A.
Voronezh, 2008
Many games use dice. The cube has 6 sides, each side has a different number of dots marked on it, from 1 to 6. The player rolls the dice and looks at how many dots there are on the dropped side (on the side that is located on top). Quite often, the points on the face of the cube are replaced with the corresponding number and then they talk about rolling out 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 occurrence of this or that event and predicting its outcome. What predictions can they make when they roll the dice? For example, these:
1) event A - the number 1, 2, 3, 4, 5 or 6 is rolled;
2) event B - the number 7, 8 or 9 appears;
3) event C - the number 1 appears.
Event A, predicted in the first case, will definitely occur. In general, an event that is sure to occur in a given experience is called reliable 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 experience is called impossible event .
And will event C, predicted in the third case, occur or not? We are not able to answer this question with complete certainty, since 1 may or may not fall out. An event that may or may not occur in a given experience is called random event .
When thinking about the occurrence of a reliable event, we most likely will not use the word “probably”. For example, if today is Wednesday, then tomorrow is Thursday, this is a reliable 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, we can say this: “With one hundred percent probability I say that tomorrow is Thursday.” On the contrary, if today is Wednesday, then the onset of Friday tomorrow is an impossible event. Assessing this event on Wednesday, we can say this: “I am sure that tomorrow is not Friday.” Or this: “It’s incredible that tomorrow is Friday.” Well, if we are prone to beautiful phrases, we can say this: “The probability that tomorrow is Friday is zero.” So, a reliable event is an event that occurs under given conditions with one hundred percent probability(i.e., occurring 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 maybe fortunately), not everything in life is so clear and precise: it will always be (certain event), it will never be (impossible event). Most often we are faced with random events, some of which are more probable, others less probable. 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 for how long percent probably a random event or how many times one random event is more likely than another. In other words, we need accurate quantitative characteristics, you need to be able to characterize probability with 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 is as zero. Given that 100% equals 1, people agreed on the following:
1) the probability of a reliable event is considered equal 1;
2) the probability of an impossible event is considered equal 0.
How to calculate the probability of a random event? After all, it happened accidentally, which means it does not obey laws, algorithms, or formulas. It turns out that in the world of randomness certain laws apply that allow one 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 event A when conducting some experiment, you should:
1) find the number N of all possible outcomes of this experiment;
2) accept the assumption of equal probability (equal possibility) of all these outcomes;
3) find the number N(A) of those experimental outcomes in which event A occurs;
4) find the quotient ; it will be equal to the probability of event A.
It is customary to denote the probability of event A: P(A). The explanation for this designation is very simple: the word “probability” in French is probabilite, in English- probability.The designation uses the first letter of the word.
Using this notation, the probability of event A according to the classical scheme can be found using the formula
P(A)=.
Often all points of the above classical probabilistic scheme are expressed in one rather long phrase.
Classic definition of probability
The probability of event A during a certain test is the ratio of the number of outcomes as a result of which 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 die the result will be: a) 4; b) 5; c) an even number of points; d) number of points greater than 4; e) number of points not divisible by three.
Solution. In total there are N=6 possible outcomes: falling out of a cube face 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 that the equiprobability of these outcomes.
a) In exactly one of the outcomes, the event A that interests us will occur—the number 4 will appear. This means that N(A)=1 and
P ( A )= =.
b) The solution and answer are the same as in the previous paragraph.
c) The event B we are interested in will occur in exactly three cases when the number of points is 2, 4 or 6. This means
N ( B )=3 and P ( B )==.
d) The event C we are interested in will occur in exactly two cases when the number of points is 5 or 6. This means
N ( C ) =2 and Р(С)=.
e) Of the six possible numbers drawn, four (1, 2, 4 and 5) are not a multiple of three, and the remaining two (3 and 6) are divisible by three. This means that the event of interest to us occurs in exactly four out of six possible and equally probable and equally probable outcomes of the experiment. Therefore the answer turns out to be
. ; b) ; V) ; G) ; d).A real dice may well differ from an ideal (model) cube, 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 greater complexity, and getting an answer becomes a problem. We limit ourselves to considering the simplest probabilistic model, where all possible outcomes are equally probable.
Note 1. Let's look at another example. The question was asked: “What is the probability of getting a three on one die roll?” The student answered: “The probability is 0.5.” And he explained his answer: “Three will either come up or not. This means that there are two outcomes in total and in exactly one of them the event of interest to us occurs. Using the classical probabilistic scheme, we get the answer 0.5.” Is there a mistake in this reasoning? At first glance, no. However, it still exists, and in a fundamental way. Yes, indeed, a three will either come up or not, i.e., with this definition of the outcome of the toss 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 implementation of point 2) is in doubt. Of course, from a purely legal point of view, we have the right to believe that rolling a three is equally likely to not falling out. But can we think so without violating our own natural assumptions about the “sameness” of the edges? Of course not! Here we are dealing with correct reasoning within a certain model. But this model itself is “wrong”, not corresponding to the real phenomenon.Note 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
, this does not mean at all that by rolling the dice 6 times you will get one point exactly once, by throwing the dice 12 times you will get one point exactly two times, by throwing the dice 18 times you will get one point exactly three times, etc. The word is probably speculative. We assume what is most likely to happen. Probably if we roll the dice 600 times, one point will come up 100 times, or about 100.