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 event is the ratio of the number of elementary outcomes favorable to 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 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 favorable to event A; - the number of all equally possible elementary outcomes of the experiment, 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 reliable event is equal to one. Let us denote a reliable event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. Let us denote an 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 for a random event the inequalities , or , are satisfied, 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 equal 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 will be blue?
Solution. We denote the event “the drawn ball turned out to be blue” by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor 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 shuffling the cards, one card is removed from the urn. What is the probability that the number on the card taken is a multiple of 5?
Solution. Let us 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 event A is favored by 6 outcomes (the numbers 5, 10, 15, 20, 25, 30). Hence, 
Example 3. Two dice are tossed and the sum of points on the top faces is calculated. Find the probability of event B such that the top faces of the dice have a total of 9 points.
Solution. In this test there are only 6 2 = 36 equally possible elementary outcomes. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), therefore 
Example 4. A natural number not greater than 10 is chosen at random. What is the probability that this number is prime?
Solution. Let us denote the event “the chosen number is prime” with the letter C. In this case, n = 10, m = 4 (prime numbers 2, 3, 5, 7). Therefore, the required probability 
Example 5. Two symmetrical coins are tossed. What is the probability that there are numbers on the top sides of both coins?
Solution. Let us denote by the letter D the event “there is a number on the top side of each coin.” In this test there are 4 equally possible elementary outcomes: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that the first coin has a coat of arms, the second one has 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 a two-digit number chosen at random has the same digits?
Solution. Two-digit numbers are numbers from 10 to 99; There are 90 such numbers in total. 9 numbers have identical digits (these are 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 identical 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?
Solution. The word differential has 12 letters, of which 5 are vowels and 7 are consonants. Letters h there is no in this word. Let us denote the events: A - “vowel letter”, B - “consonant letter”, C - “letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n = 12, then
, And .
Example 8. Two dice are tossed and the number of points on the top of each dice is noted. Find the probability that both dice show the same number of points.
Solution. Let's 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). The total number of equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. This means that the required probability 
Example 9. The book has 300 pages. What is the probability that a randomly opened page will have a serial number divisible by 5?
Solution. From the conditions of the problem it follows that all equally possible elementary outcomes that form a complete group of events will be n = 300. 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 tossed and the sum of points on the top faces is calculated. What is more likely - getting a total of 7 or 8?
Solution. Let us denote the events: A - “7 points are rolled”, B – “8 points are rolled”. Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B is favored by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). All equally possible elementary outcomes are n = 6 2 = 36. This means 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, identical in size and weight. What is the probability that a ball drawn at random from this urn will be blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of 30?
4. In the urn A blue and b red balls, identical in size and weight. One ball is taken from this urn and set aside. This ball turned out to be red. After this, another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A national number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are tossed and the sum of points on the top faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed and the sum of the points rolled 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 = 25/216 - probability of getting 9 points in total; p 2 = 27/216 - probability of getting 10 points in total; p 2 > p 1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).
Questions
1. What is the probability of an event called?
2. What is the probability of a reliable 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?
- Probability is the degree (relative measure, quantitative assessment) of the possibility of the occurrence of some event. When the reasons for some possible event to actually occur outweigh the opposing reasons, then this event is called probable, otherwise - unlikely or improbable. The preponderance of positive reasons over negative ones, and vice versa, can be to varying degrees, as a result of which the probability (and improbability) can be greater or lesser. Therefore, probability is often assessed at a qualitative level, especially in cases where a more or less accurate quantitative assessment is impossible or extremely difficult. Various gradations of “levels” of probability are possible.
The study of probability from a mathematical point of view constitutes a special discipline - probability theory. In probability theory and mathematical statistics, the concept of probability is formalized as a numerical characteristic of an event - a probability measure (or its value) - a measure on a set of events (subsets of a set of elementary events), taking values from
(\displaystyle 0)
(\displaystyle 1)
Meaning
(\displaystyle 1)
Corresponds to a reliable event. An impossible event has a probability of 0 (the converse is generally not always true). If the probability of an event occurring is
(\displaystyle p)
Then the probability of its non-occurrence is equal to
(\displaystyle 1-p)
In particular, the probability
(\displaystyle 1/2)
Means equal probability of occurrence and non-occurrence of an event.
The classical definition of probability is based on the concept of equal probability of outcomes. The probability is the ratio of the number of outcomes favorable for a given event to the total number of equally possible outcomes. For example, the probability of getting heads or tails in a random coin toss is 1/2 if it is assumed that only these two possibilities occur and that they are equally possible. This classical “definition” of probability can be generalized to the case of an infinite number of possible values - for example, if some event can occur with equal probability at any point (the number of points is infinite) of some limited region of space (plane), then the probability that it will occur in some part of this feasible region is equal to the ratio of the volume (area) of this part to the volume (area) of the region of all possible points.
The empirical “definition” of probability is related to the frequency of an event, based on the fact that with a sufficiently large number of trials, the frequency should tend to the objective degree of possibility of this event. In the modern presentation of probability theory, probability is defined axiomatically, as a special case of the abstract theory of set measure. However, the connecting link between the abstract measure and the probability, which expresses the degree of possibility of the occurrence of an event, is precisely the frequency of its observation.
The probabilistic description of certain phenomena has become widespread in modern science, in particular in econometrics, statistical physics of macroscopic (thermodynamic) systems, where even in the case of a classical deterministic description of the movement of particles, a deterministic description of the entire system of particles does not seem practically possible or appropriate. In quantum physics, the processes described are themselves probabilistic in nature.
Presented to date in the open bank of Unified State Exam problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is the classical definition of probability.
The easiest way to understand the formula is with examples.
Example 1. There are 9 red balls and 3 blue balls in the basket. The balls differ only in color. We take out one of them at random (without looking). What is the probability that the ball chosen in this way will be blue?
A comment. In problems in probability theory, something happens (in this case, our action of pulling out the ball) that can have a different result - an outcome. It should be noted that the result can be looked at in different ways. “We pulled out some kind of ball” is also a result. “We pulled out the blue ball” - the result. “We pulled out exactly this ball from all possible balls” - this least generalized view of the result is called an elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.
Solution. Now let's calculate the probability of choosing the blue ball.
Event A: “the selected ball turned out to be blue”
Total number of all possible outcomes: 9+3=12 (the number of all balls that we could draw)
Number of outcomes favorable for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P(A)=3/12=1/4=0.25
Answer: 0.25
For the same problem, let's calculate the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. Number of favorable outcomes: 9. Probability sought: 9/12=3/4=0.75
The probability of any event always lies between 0 and 1.
Sometimes in everyday speech (but not in probability theory!) the probability of events is estimated as a percentage. The transition between math and conversational scores is accomplished by multiplying (or dividing) by 100%.
So,
Moreover, the probability is zero for events that cannot happen - incredible. For example, in our example this would be the probability of drawing a green ball from the basket. (The number of favorable outcomes is 0, P(A)=0/12=0, if calculated using the formula)
Probability 1 has events that are absolutely certain to happen, without options. For example, the probability that “the selected ball will be either red or blue” is for our task. (Number of favorable outcomes: 12, P(A)=12/12=1)
We looked at a classic example illustrating the definition of probability. All similar problems of the Unified State Exam in probability theory are solved by using this formula.
In place of the red and blue balls there may be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on some topic (prototypes,), defective and high-quality bags or garden pumps (prototypes,) - the principle remains the same.
They differ slightly in the formulation of the problem of the probability theory of the Unified State Examination, where you need to calculate the probability of some event occurring on a certain day. ( , ) As in previous problems, you need to determine what is the elementary outcome, and then apply the same formula.
Example 2. The conference lasts three days. On the first and second days there are 15 speakers, on the third day - 20. What is the probability that Professor M.’s report will fall on the third day if the order of reports is determined by drawing lots?
What is the elementary outcome here? – Assigning the professor’s report one of all possible serial numbers for the speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report may receive one of 50 issues. This means there are only 50 elementary outcomes.
What are the favorable outcomes? - Those in which it turns out that the professor will speak on the third day. That is, the last 20 numbers.
According to the formula, probability P(A)= 20/50=2/5=4/10=0.4
Answer: 0.4
The drawing of lots here represents the establishment of a random correspondence between people and ordered places. In example 2, matching was considered from the point of view of which of the places a particular person could occupy. You can approach the same situation from the other side: which of the people with what probability could get to a specific place (prototypes , , , ):
Example 3. The draw includes 5 Germans, 8 French and 3 Estonians. What is the probability that the first (/second/seventh/last – it doesn’t matter) will be a Frenchman.
The number of elementary outcomes is the number of all possible people who could get into a given place by drawing lots. 5+8+3=16 people.
Favorable outcomes - French. 8 people.
Required probability: 8/16=1/2=0.5
Answer: 0.5
The prototype is slightly different. There are still problems about coins () and dice (), which are somewhat more creative. The solution to these problems can be found on the prototype pages.
Here are a few examples of tossing a coin or dice.
Example 4. When we toss a coin, what is the probability of landing on heads?
There are 2 outcomes – heads or tails. (it is believed that the coin never lands on its edge) A favorable outcome is tails, 1.
Probability 1/2=0.5
Answer: 0.5.
Example 5. What if we toss a coin twice? What is the probability of getting heads both times?
The main thing is to determine what elementary outcomes we will consider when tossing two coins. After tossing two coins, one of the following results can occur:
1) PP – both times it came up heads
2) PO – first time heads, second time heads
3) OP – heads the first time, tails the second time
4) OO – heads came up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one, 1, is favorable.
Probability: 1/4=0.25
Answer: 0.25
What is the probability that two coin tosses will result in tails?
The number of elementary outcomes is the same, 4. Favorable outcomes are the second and third, 2.
Probability of getting one tail: 2/4=0.5
In such problems, another formula may be useful.
If with one toss of a coin we have 2 possible outcome options, then for two tosses the results will be 2 2 = 2 2 = 4 (as in example 5), for three tosses 2 2 2 = 2 3 = 8, for four: 2·2·2·2=2 4 =16, ... for N rolls the possible results will be 2·2·...·2=2 N .
So, you can find the probability of getting 5 heads out of 5 coin tosses.
Total number of elementary outcomes: 2 5 =32.
Favorable outcomes: 1. (RRRRRR – heads all 5 times)
Probability: 1/32=0.03125
The same is true for dice. With one throw, there are 6 possible results. So, for two throws: 6 6 = 36, for three 6 6 6 = 216, etc.
Example 6. We throw the dice. What is the probability that an even number will be rolled?
Total outcomes: 6, according to the number of sides.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6=0.5
Example 7. We throw two dice. What is the probability that the total will be 10? (round to the nearest hundredth)
For one die there are 6 possible outcomes. This means that for two, according to the above rule, 6·6=36.
What outcomes will be favorable for the total to roll 10?
10 must be decomposed into the sum of two numbers from 1 to 6. This can be done in two ways: 10=6+4 and 10=5+5. This means that the following options are possible for the cubes:
(6 on the first and 4 on the second)
(4 on the first and 6 on the second)
(5 on the first and 5 on the second)
Total, 3 options. Required probability: 3/36=1/12=0.08
Answer: 0.08
Other types of B6 problems will be discussed in a future How to Solve article.
Initially, being just a collection of information and empirical observations about the game of dice, the theory of probability became a thorough science. The first to give it a mathematical framework were Fermat and Pascal.
From thinking about the eternal to the theory of probability
The two individuals to whom probability theory owes many of its fundamental formulas, Blaise Pascal and Thomas Bayes, are known as deeply religious people, the latter being a Presbyterian minister. Apparently, the desire of these two scientists to prove the fallacy of the opinion about a certain Fortune, who bestows good luck on her favorites, gave impetus to research in this area. After all, in fact, any gambling game with its winnings and losses is just a symphony of mathematical principles.
Thanks to the passion of the Chevalier de Mere, who was equally a gambler and a man not indifferent to science, Pascal was forced to find a way to calculate probability. De Mere was interested in the following question: “How many times do you need to throw two dice in pairs so that the probability of getting 12 points exceeds 50%?” The second question, which was of great interest to the gentleman: “How to divide the bet between the participants in the unfinished game?” Of course, Pascal successfully answered both questions of de Mere, who became the unwitting initiator of the development of probability theory. It is interesting that the person of de Mere remained known in this area, and not in literature.
Previously, no mathematician had ever attempted to calculate the probabilities of events, since it was believed that this was only a guessing solution. Blaise Pascal gave the first definition of the probability of an event and showed that it is a specific figure that can be justified mathematically. Probability theory has become the basis for statistics and is widely used in modern science.
What is randomness
If we consider a test that can be repeated an infinite number of times, then we can define a random event. This is one of the likely outcomes of the experiment.
Experience is the implementation of specific actions under constant conditions.
To be able to work with the results of the experiment, events are usually designated by the letters A, B, C, D, E...
Probability of a random event
In order to begin the mathematical part of probability, it is necessary to define all its components.
The probability of an event is a numerical measure of the possibility of some event (A or B) occurring as a result of an experience. The probability is denoted as P(A) or P(B).
In probability theory they distinguish:
- reliable the event is guaranteed to occur as a result of the experience P(Ω) = 1;
- impossible the event can never happen P(Ø) = 0;
- random an event lies between reliable and impossible, that is, the probability of its occurrence is possible, but not guaranteed (the probability of a random event is always within the range 0≤Р(А)≤ 1).
Relationships between events
Both one and the sum of events A+B are considered, when the event is counted when at least one of the components, A or B, or both, A and B, is fulfilled.
In relation to each other, events can be:
- Equally possible.
- Compatible.
- Incompatible.
- Opposite (mutually exclusive).
- Dependent.
If two events can happen with equal probability, then they equally possible.
If the occurrence of event A does not reduce to zero the probability of the occurrence of event B, then they compatible.
If events A and B never occur simultaneously in the same experience, then they are called incompatible. Tossing a coin is a good example: the appearance of heads is automatically the non-appearance of heads.
The probability for the sum of such incompatible events consists of the sum of the probabilities of each of the events:
P(A+B)=P(A)+P(B)
If the occurrence of one event makes the occurrence of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as “not A”). The occurrence of event A means that Ā did not occur. These two events form a complete group with a sum of probabilities equal to 1.
Dependent events have mutual influence, decreasing or increasing the probability of each other.
Relationships between events. Examples
Using examples it is much easier to understand the principles of probability theory and combinations of events.
The experiment that will be carried out consists of taking balls out of a box, and the result of each experiment is an elementary outcome.
An event is one of the possible outcomes of an experiment - a red ball, a blue ball, a ball with number six, etc.
Test No. 1. There are 6 balls involved, three of which are blue with odd numbers on them, and the other three are red with even numbers.
Test No. 2. There are 6 blue balls with numbers from one to six.
Based on this example, we can name combinations:
- Reliable event. In Spanish No. 2 the event “get the blue ball” is reliable, since the probability of its occurrence is equal to 1, since all the balls are blue and there can be no miss. Whereas the event “get the ball with the number 1” is random.
- Impossible event. In Spanish No. 1 with blue and red balls, the event “getting the purple ball” is impossible, since the probability of its occurrence is 0.
- Equally possible events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally possible, and the events “get the ball with an even number” and “get the ball with the number 2” have different probabilities.
- Compatible Events. Getting a six twice in a row while throwing a die is a compatible event.
- Incompatible events. In the same Spanish No. 1, the events “get a red ball” and “get a ball with an odd number” cannot be combined in the same experience.
- Opposite events. The most striking example of this is coin tossing, where drawing heads is equivalent to not drawing tails, and the sum of their probabilities is always 1 (full group).
- Dependent Events. So, in Spanish No. 1, you can set the goal of drawing the red ball twice in a row. Whether or not it is retrieved the first time affects the likelihood of being retrieved the second time.
It can be seen that the first event significantly affects the probability of the second (40% and 60%).
Event probability formula
The transition from fortune-telling to precise data occurs through the translation of the topic into a mathematical plane. That is, judgments about a random event such as “high probability” or “minimal probability” can be translated into specific numerical data. It is already permissible to evaluate, compare and enter such material into more complex calculations.
From a calculation point of view, determining the probability of an event is the ratio of the number of elementary positive outcomes to the number of all possible outcomes of experience regarding a specific event. Probability is denoted by P(A), where P stands for the word “probabilite”, which is translated from French as “probability”.
So, the formula for the probability of an event is:
Where m is the number of favorable outcomes for event A, n is the sum of all outcomes possible for this experience. In this case, the probability of an event always lies between 0 and 1:
0 ≤ P(A)≤ 1.
Calculation of the probability of an event. Example
Let's take Spanish. No. 1 with balls, which was described earlier: 3 blue balls with the numbers 1/3/5 and 3 red balls with the numbers 2/4/6.
Based on this test, several different problems can be considered:
- A - red ball falling out. There are 3 red balls, and there are 6 options in total. This is the simplest example in which the probability of an event is P(A)=3/6=0.5.
- B - rolling an even number. There are 3 even numbers (2,4,6), and the total number of possible numerical options is 6. The probability of this event is P(B)=3/6=0.5.
- C - the occurrence of a number greater than 2. There are 4 such options (3,4,5,6) out of a total number of possible outcomes of 6. The probability of event C is equal to P(C)=4/6=0.67.
As can be seen from the calculations, event C has a higher probability, since the number of probable positive outcomes is higher than in A and B.
Incompatible events
Such events cannot appear simultaneously in the same experience. As in Spanish No. 1 it is impossible to get a blue and a red ball at the same time. That is, you can get either a blue or a red ball. In the same way, an even and an odd number cannot appear in a dice at the same time.
The probability of two events is considered as the probability of their sum or product. The sum of such events A+B is considered to be an event that consists of the occurrence of event A or B, and the product of them AB is the occurrence of both. For example, the appearance of two sixes at once on the faces of two dice in one throw.
The sum of several events is an event that presupposes the occurrence of at least one of them. The production of several events is the joint occurrence of them all.
In probability theory, as a rule, the use of the conjunction “and” denotes a sum, and the conjunction “or” - multiplication. Formulas with examples will help you understand the logic of addition and multiplication in probability theory.
Probability of the sum of incompatible events
If the probability of incompatible events is considered, then the probability of the sum of events is equal to the addition of their probabilities:
P(A+B)=P(A)+P(B)
For example: let's calculate the probability that in Spanish. No. 1 with blue and red balls, a number between 1 and 4 will appear. We will calculate not in one action, but by the sum of the probabilities of the elementary components. So, in such an experiment there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting the number 2 is 1/6, the probability of getting the number 3 is also 1/6. The probability of getting a number between 1 and 4 is:
The probability of the sum of incompatible events of a complete group is 1.
So, if in an experiment with a cube we add up the probabilities of all numbers appearing, the result will be one.
This is also true for opposite events, for example in the experiment with a coin, where one side is the event A, and the other is the opposite event Ā, as is known,
P(A) + P(Ā) = 1
Probability of incompatible events occurring
Probability multiplication is used when considering the occurrence of two or more incompatible events in one observation. The probability that events A and B will appear in it simultaneously is equal to the product of their probabilities, or:
P(A*B)=P(A)*P(B)
For example, the probability that in Spanish No. 1, as a result of two attempts, a blue ball will appear twice, equal to
That is, the probability of an event occurring when, as a result of two attempts to extract balls, only blue balls are extracted is 25%. It is very easy to do practical experiments on this problem and see if this is actually the case.
Joint events
Events are considered joint when the occurrence of one of them can coincide with the occurrence of another. Despite the fact that they are joint, the probability of independent events is considered. For example, throwing two dice can give a result when the number 6 appears on both of them. Although the events coincided and appeared at the same time, they are independent of each other - only one six could fall out, the second die has no influence on it.
The probability of joint events is considered as the probability of their sum.
Probability of the sum of joint events. Example
The probability of the sum of events A and B, which are joint in relation to each other, is equal to the sum of the probabilities of the event minus the probability of their occurrence (that is, their joint occurrence):
R joint (A+B)=P(A)+P(B)- P(AB)
Let's assume that the probability of hitting the target with one shot is 0.4. Then event A is hitting the target in the first attempt, B - in the second. These events are joint, since it is possible that you can hit the target with both the first and second shots. But events are not dependent. What is the probability of the event of hitting the target with two shots (at least with one)? According to the formula:
0,4+0,4-0,4*0,4=0,64
The answer to the question is: “The probability of hitting the target with two shots is 64%.”
This formula for the probability of an event can also be applied to incompatible events, where the probability of the joint occurrence of an event P(AB) = 0. This means that the probability of the sum of incompatible events can be considered a special case of the proposed formula.
Geometry of probability for clarity
Interestingly, the probability of the sum of joint events can be represented as two areas A and B, which intersect with each other. As can be seen from the picture, the area of their union is equal to the total area minus the area of their intersection. This geometric explanation makes the seemingly illogical formula more understandable. Note that geometric solutions are not uncommon in probability theory.
Determining the probability of the sum of many (more than two) joint events is quite cumbersome. To calculate it, you need to use the formulas that are provided for these cases.
Dependent Events
Events are called dependent if the occurrence of one (A) of them affects the probability of the occurrence of another (B). Moreover, the influence of both the occurrence of event A and its non-occurrence is taken into account. Although events are called dependent by definition, only one of them is dependent (B). Ordinary probability was denoted as P(B) or the probability of independent events. In the case of dependent events, a new concept is introduced - conditional probability P A (B), which is the probability of a dependent event B, subject to the occurrence of event A (hypothesis), on which it depends.
But event A is also random, so it also has a probability that needs and can be taken into account in the calculations performed. The following example will show how to work with dependent events and a hypothesis.
An example of calculating the probability of dependent events
A good example for calculating dependent events would be a standard deck of cards.
Using a deck of 36 cards as an example, let’s look at dependent events. We need to determine the probability that the second card drawn from the deck will be of diamonds if the first card drawn is:
- Bubnovaya.
- A different color.
Obviously, the probability of the second event B depends on the first A. So, if the first option is true, that there is 1 card (35) and 1 diamond (8) less in the deck, the probability of event B:
R A (B) =8/35=0.23
If the second option is true, then the deck has 35 cards, and the full number of diamonds (9) is still retained, then the probability of the following event B:
R A (B) =9/35=0.26.
It can be seen that if event A is conditioned on the fact that the first card is a diamond, then the probability of event B decreases, and vice versa.
Multiplying dependent events
Guided by the previous chapter, we accept the first event (A) as a fact, but in essence, it is of a random nature. The probability of this event, namely drawing a diamond from a deck of cards, is equal to:
P(A) = 9/36=1/4
Since the theory does not exist on its own, but is intended to serve for practical purposes, it is fair to note that what is most often needed is the probability of producing dependent events.
According to the theorem on the product of probabilities of dependent events, the probability of occurrence of jointly dependent events A and B is equal to the probability of one event A, multiplied by the conditional probability of event B (dependent on A):
P(AB) = P(A) *P A(B)
Then, in the deck example, the probability of drawing two cards with the suit of diamonds is:
9/36*8/35=0.0571, or 5.7%
And the probability of extracting not diamonds first, and then diamonds, is equal to:
27/36*9/35=0.19, or 19%
It can be seen that the probability of event B occurring is greater provided that the first card drawn is of a suit other than diamonds. This result is quite logical and understandable.
Total probability of an event
When a problem with conditional probabilities becomes multifaceted, it cannot be calculated using conventional methods. When there are more than two hypotheses, namely A1, A2,…, A n, ..forms a complete group of events provided:
- P(A i)>0, i=1,2,…
- A i ∩ A j =Ø,i≠j.
- Σ k A k =Ω.
So, the formula for the total probability for event B with a complete group of random events A1, A2,..., A n is equal to:
A look into the future
The probability of a random event is extremely necessary in many areas of science: econometrics, statistics, physics, etc. Since some processes cannot be described deterministically, since they themselves are probabilistic in nature, special working methods are required. The theory of event probability can be used in any technological field as a way to determine the possibility of an error or malfunction.
We can say that by recognizing probability, we in some way take a theoretical step into the future, looking at it through the prism of formulas.