How to determine the probability of an event formula. Probability theory

Many, when faced with the concept of “probability theory,” get scared, thinking that it is something overwhelming, very complex. But everything is actually not so tragic. Today we will look at the basic concept of probability theory and learn how to solve problems using specific examples.

The science

What does such a branch of mathematics as “probability theory” study? She notes patterns and quantities. Scientists first became interested in this issue back in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. It is any fact that is established by experience or observation. But what is experience? Another basic concept of probability theory. It means that this set of circumstances was created not by chance, but for a specific purpose. As for observation, here the researcher himself does not participate in the experiment, but is simply a witness to these events; he does not influence what is happening in any way.

Events

We learned that the basic concept of probability theory is an event, but we did not consider the classification. All of them are divided into the following categories:

Reliable.
Impossible.
Random.

Regardless of what kind of events they are, observed or created during the experience, they are all subject to this classification. We invite you to get acquainted with each type separately.

Reliable event

This is a circumstance for which the necessary set of measures has been taken. In order to better understand the essence, it is better to give a few examples. Physics, chemistry, economics, and higher mathematics are subject to this law. The theory of probability includes such an important concept as a reliable event. Here are some examples:

We work and receive compensation in the form of wages.
We passed the exams well, passed the competition, and for this we receive a reward in the form of admission to an educational institution.
We invested money in the bank, and if necessary, we will get it back.

Such events are reliable. If we have fulfilled all the necessary conditions, we will definitely get the expected result.

Impossible events

Now we are considering elements of probability theory. We propose to move on to an explanation of the next type of event, namely the impossible. First, let's stipulate the most important rule - the probability of an impossible event is zero.

One cannot deviate from this formulation when solving problems. For clarification, here are examples of such events:

The water froze at a temperature of plus ten (this is impossible).
The lack of electricity does not affect production in any way (just as impossible as in the previous example).

It is not worth giving more examples, since those described above very clearly reflect the essence of this category. An impossible event will never occur during an experiment under any circumstances.

Random Events

When studying the elements, special attention should be paid to this particular type of event. This is what science studies. As a result of the experience, something may or may not happen. In addition, the test can be carried out an unlimited number of times. Vivid examples include:

The toss of a coin is an experience or test, the landing of heads is an event.
Pulling a ball out of a bag blindly is a test; getting a red ball is an event, and so on.

There can be an unlimited number of such examples, but, in general, the essence should be clear. To summarize and systematize the knowledge gained about the events, a table is provided. Probability theory studies only the last type of all presented.

Name	definition
Reliable	Events that occur with a 100% guarantee if certain conditions are met.	Admission to an educational institution upon passing the entrance exam well.
Impossible	Events that will never happen under any circumstances.	It is snowing at an air temperature of plus thirty degrees Celsius.
Random	An event that may or may not occur during an experiment/test.	A hit or miss when throwing a basketball into a hoop.

Laws

Probability theory is a science that studies the possibility of an event occurring. Like the others, it has some rules. The following laws of probability theory exist:

Convergence of sequences of random variables.
Law of large numbers.

When calculating the possibility of something complex, you can use a set of simple events to achieve a result in an easier and faster way. Note that the laws of probability theory are easily proven using certain theorems. We suggest that you first get acquainted with the first law.

Convergence of sequences of random variables

Note that there are several types of convergence:

The sequence of random variables converges in probability.
Almost impossible.
Mean square convergence.
Distribution convergence.

So, right off the bat, it’s very difficult to understand the essence. Here are definitions that will help you understand this topic. Let's start with the first view. The sequence is called convergent in probability, if the following condition is met: n tends to infinity, the number to which the sequence tends is greater than zero and close to one.

Let's move on to the next view, almost certainly. The sequence is said to converge almost certainly to a random variable with n tending to infinity and P tending to a value close to unity.

The next type is mean square convergence. When using SC convergence, the study of vector random processes is reduced to the study of their coordinate random processes.

The last type remains, let's look at it briefly so that we can move directly to solving problems. Convergence in distribution has another name - “weak”, and we will explain why later. Weak convergence is the convergence of distribution functions at all points of continuity of the limiting distribution function.

We will definitely keep our promise: weak convergence differs from all of the above in that the random variable is not defined in the probability space. This is possible because the condition is formed exclusively using distribution functions.

Law of Large Numbers

Theorems of probability theory, such as:

Chebyshev's inequality.
Chebyshev's theorem.
Generalized Chebyshev's theorem.
Markov's theorem.

If we consider all these theorems, then this question may drag on for several dozen sheets. Our main task is to apply probability theory in practice. We suggest you do this right now. But before that, let’s look at the axioms of probability theory; they will be the main assistants in solving problems.

Axioms

We already met the first one when we talked about an impossible event. Let's remember: the probability of an impossible event is zero. We gave a very vivid and memorable example: snow fell at an air temperature of thirty degrees Celsius.

The second is as follows: a reliable event occurs with a probability equal to one. Now we will show how to write this using mathematical language: P(B)=1.

Third: A random event may or may not happen, but the possibility always ranges from zero to one. The closer the value is to one, the greater the chances; if the value approaches zero, the probability is very low. Let's write this in mathematical language: 0<Р(С)<1.

Let's consider the last, fourth axiom, which sounds like this: the probability of the sum of two events is equal to the sum of their probabilities. We write it in mathematical language: P(A+B)=P(A)+P(B).

The axioms of probability theory are the simplest rules that are not difficult to remember. Let's try to solve some problems based on the knowledge we have already acquired.

Lottery ticket

First, let's look at the simplest example - a lottery. Imagine that you bought one lottery ticket for good luck. What is the probability that you will win at least twenty rubles? In total, a thousand tickets are participating in the circulation, one of which has a prize of five hundred rubles, ten of them have a hundred rubles each, fifty have a prize of twenty rubles, and one hundred have a prize of five. Probability problems are based on finding the possibility of luck. Now together we will analyze the solution to the above task.

If we use the letter A to denote a win of five hundred rubles, then the probability of getting A will be equal to 0.001. How did we get this? You just need to divide the number of “lucky” tickets by their total number (in this case: 1/1000).

B is a win of one hundred rubles, the probability will be 0.01. Now we acted on the same principle as in the previous action (10/1000)

C - the winnings are twenty rubles. We find the probability, it is equal to 0.05.

We are not interested in the remaining tickets, since their prize fund is less than that specified in the condition. Let's apply the fourth axiom: The probability of winning at least twenty rubles is P(A)+P(B)+P(C). The letter P denotes the probability of the occurrence of a given event; we have already found them in previous actions. All that remains is to add up the necessary data, and the answer we get is 0.061. This number will be the answer to the task question.

Card deck

Problems in probability theory can be more complex; for example, let’s take the following task. In front of you is a deck of thirty-six cards. Your task is to draw two cards in a row without shuffling the stack, the first and second cards must be aces, the suit does not matter.

First, let's find the probability that the first card will be an ace, for this we divide four by thirty-six. They put it aside. We take out the second card, it will be an ace with a probability of three thirty-fifths. The probability of the second event depends on which card we drew first, we wonder whether it was an ace or not. It follows from this that event B depends on event A.

The next step is to find the probability of simultaneous occurrence, that is, we multiply A and B. Their product is found as follows: we multiply the probability of one event by the conditional probability of another, which we calculate, assuming that the first event occurred, that is, we drew an ace with the first card.

To make everything clear, let’s give a designation to such an element as events. It is calculated assuming that event A has occurred. It is calculated as follows: P(B/A).

Let's continue solving our problem: P(A * B) = P(A) * P(B/A) or P(A * B) = P(B) * P(A/B). The probability is equal to (4/36) * ((3/35)/(4/36). We calculate by rounding to the nearest hundredth. We have: 0.11 * (0.09/0.11) = 0.11 * 0, 82 = 0.09. The probability that we will draw two aces in a row is nine hundredths. The value is very small, it follows that the probability of the event occurring is extremely small.

Forgotten number

We propose to analyze several more variants of tasks that are studied by probability theory. You have already seen examples of solving some of them in this article. Let’s try to solve the following problem: the boy forgot the last digit of his friend’s phone number, but since the call was very important, he began to dial everything one by one. We need to calculate the probability that he will call no more than three times. The solution to the problem is simplest if the rules, laws and axioms of probability theory are known.

Before looking at the solution, try solving it yourself. We know that the last digit can be from zero to nine, that is, ten values in total. The probability of getting the right one is 1/10.

Next, we need to consider the options for the origin of the event, suppose that the boy guessed right and immediately typed the right one, the probability of such an event is 1/10. Second option: the first call misses, and the second one is on target. Let's calculate the probability of such an event: multiply 9/10 by 1/9, and as a result we also get 1/10. The third option: the first and second calls turned out to be at the wrong address, only with the third the boy got to where he wanted. We calculate the probability of such an event: 9/10 multiplied by 8/9 and 1/8, resulting in 1/10. We are not interested in other options according to the conditions of the problem, so we just have to add up the results obtained, in the end we have 3/10. Answer: the probability that the boy will call no more than three times is 0.3.

Cards with numbers

There are nine cards in front of you, on each of which a number from one to nine is written, the numbers are not repeated. They were put in a box and mixed thoroughly. You need to calculate the probability that

an even number will appear;
two-digit.

Before moving on to the solution, let's stipulate that m is the number of successful cases, and n is the total number of options. Let's find the probability that the number will be even. It won’t be difficult to calculate that there are four even numbers, this will be our m, there are nine possible options in total, that is, m=9. Then the probability is 0.44 or 4/9.

Let's consider the second case: the number of options is nine, and there can be no successful outcomes at all, that is, m equals zero. The probability that the drawn card will contain a two-digit number is also zero.

When a coin is tossed, we can say that it will land heads up, or probability this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land on heads 5 times. If the coin is "fair" and if it is tossed many times, then heads will land very close half the time. Thus, there are two types of probabilities: experimental And theoretical .

Experimental and theoretical probability

If we flip a coin a large number of times - say 1000 - and count how many times it lands on heads, we can determine the probability that it lands on heads. If heads are thrown 503 times, we can calculate the probability of it landing:
503/1000, or 0.503.

This experimental determination of probability. This definition of probability comes from observation and study of data and is quite common and very useful. Here, for example, are some probabilities that were determined experimentally:

1. The probability that a woman will develop breast cancer is 1/11.

2. If you kiss someone who has a cold, then the probability that you will also get a cold is 0.07.

3. A person who has just been released from prison has an 80% chance of returning to prison.

If we consider tossing a coin and taking into account that it is just as likely that it will come up heads or tails, we can calculate the probability of getting heads: 1/2. This is a theoretical definition of probability. Here are some other probabilities that have been determined theoretically using mathematics:

1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding year) is 0.706.

2. During a trip, you meet someone, and during the conversation you discover that you have a mutual friend. Typical reaction: “This can’t be!” In fact, this phrase is not suitable, because the probability of such an event is quite high - just over 22%.

Thus, experimental probabilities are determined through observation and data collection. Theoretical probabilities are determined through mathematical reasoning. Examples of experimental and theoretical probabilities, such as those discussed above, and especially those that we do not expect, lead us to the importance of studying probability. You may ask, "What is true probability?" In fact, there is no such thing. Probabilities within certain limits can be determined experimentally. They may or may not coincide with the probabilities that we obtain theoretically. There are situations in which it is much easier to determine one type of probability than another. For example, it would be sufficient to find the probability of catching a cold using theoretical probability.

Calculation of experimental probabilities

Let us first consider the experimental definition of probability. The basic principle we use to calculate such probabilities is as follows.

Principle P (experimental)

If in an experiment in which n observations are made, a situation or event E occurs m times in n observations, then the experimental probability of the event is said to be P (E) = m/n.

Example 1 Sociological survey. An experimental study was conducted to determine the number of left-handed people, right-handed people and people whose both hands are equally developed. The results are shown in the graph.

a) Determine the probability that the person is right-handed.

b) Determine the probability that the person is left-handed.

c) Determine the probability that a person is equally fluent in both hands.

d) Most Professional Bowling Association tournaments are limited to 120 players. Based on the data from this experiment, how many players could be left-handed?

Solution

a)The number of people who are right-handed is 82, the number of left-handers is 17, and the number of those who are equally fluent in both hands is 1. The total number of observations is 100. Thus, the probability that a person is right-handed is P
P = 82/100, or 0.82, or 82%.

b) The probability that a person is left-handed is P, where
P = 17/100, or 0.17, or 17%.

c) The probability that a person is equally fluent in both hands is P, where
P = 1/100, or 0.01, or 1%.

d) 120 bowlers, and from (b) we can expect that 17% are left-handed. From here
17% of 120 = 0.17.120 = 20.4,
that is, we can expect about 20 players to be left-handed.

Example 2 Quality control . It is very important for a manufacturer to keep the quality of its products at a high level. In fact, companies hire quality control inspectors to ensure this process. The goal is to produce the minimum possible number of defective products. But since the company produces thousands of products every day, it cannot afford to test every product to determine whether it is defective or not. To find out what percentage of products are defective, the company tests far fewer products.
The USDA requires that 80% of the seeds sold by growers must germinate. To determine the quality of the seeds that an agricultural company produces, 500 seeds from those that were produced are planted. After this, it was calculated that 417 seeds sprouted.

a) What is the probability that the seed will germinate?

b) Do the seeds meet government standards?

Solution a) We know that out of 500 seeds that were planted, 417 sprouted. Probability of seed germination P, and
P = 417/500 = 0.834, or 83.4%.

b) Since the percentage of seeds germinated has exceeded 80% as required, the seeds meet government standards.

Example 3 Television ratings. According to statistics, there are 105,500,000 households with televisions in the United States. Every week, information about viewing programs is collected and processed. In one week, 7,815,000 households tuned in to the hit comedy series "Everybody Loves Raymond" on CBS and 8,302,000 households tuned in to the hit series "Law & Order" on NBC (Source: Nielsen Media Research). What is the probability that one household's TV is tuned to "Everybody Loves Raymond" during a given week? to "Law & Order"?

Solution The probability that the TV in one household is tuned to "Everybody Loves Raymond" is P, and
P = 7,815,000/105,500,000 ≈ 0.074 ≈ 7.4%.
The chance that the household's TV was tuned to Law & Order is P, and
P = 8,302,000/105,500,000 ≈ 0.079 ≈ 7.9%.
These percentages are called ratings.

Theoretical probability

Suppose we are conducting an experiment, such as throwing a coin or darts, drawing a card from a deck, or testing products for quality on an assembly line. Each possible result of such an experiment is called Exodus . The set of all possible outcomes is called outcome space . Event it is a set of outcomes, that is, a subset of the space of outcomes.

Example 4 Throwing darts. Suppose that in a dart throwing experiment, a dart hits a target. Find each of the following:

b) Outcome space

Solution
a) The outcomes are: hitting black (B), hitting red (R) and hitting white (B).

b) The space of outcomes is (hitting black, hitting red, hitting white), which can be written simply as (H, K, B).

Example 5 Throwing dice. A die is a cube with six sides, each with one to six dots on it.

Suppose we are throwing a die. Find
a) Outcomes
b) Outcome space

Solution
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Outcome space (1, 2, 3, 4, 5, 6).

We denote the probability that an event E occurs as P(E). For example, “the coin will land on heads” can be denoted by H. Then P(H) represents the probability that the coin will land on heads. When all outcomes of an experiment have the same probability of occurring, they are said to be equally likely. To see the differences between events that are equally likely and events that are not, consider the target shown below.

For target A, the events of hitting black, red and white are equally probable, since the black, red and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, hitting them is not equally probable.

Principle P (Theoretical)

If an event E can happen in m ways out of n possible equally probable outcomes from the outcome space S, then theoretical probability events, P(E) is
P(E) = m/n.

Example 6 What is the probability of rolling a die to get a 3?

Solution There are 6 equally probable outcomes on a dice and there is only one possibility of rolling the number 3. Then the probability P will be P(3) = 1/6.

Example 7 What is the probability of rolling an even number on a die?

Solution The event is the throwing of an even number. This can happen in 3 ways (if you roll a 2, 4 or 6). The number of equally probable outcomes is 6. Then the probability P(even) = 3/6, or 1/2.

We will use a number of examples involving a standard 52 card deck. This deck consists of the cards shown in the figure below.

Example 8 What is the probability of drawing an Ace from a well-shuffled deck of cards?

Solution There are 52 outcomes (the number of cards in the deck), they are equally likely (if the deck is well shuffled), and there are 4 ways to draw an Ace, so according to the P principle, the probability
P(draw an ace) = 4/52, or 1/13.

Example 9 Suppose we choose, without looking, one ball from a bag with 3 red balls and 4 green balls. What is the probability of choosing a red ball?

Solution There are 7 equally probable outcomes of drawing any ball, and since the number of ways to draw a red ball is 3, we get
P(red ball selection) = 3/7.

The following statements are results from Principle P.

Properties of Probability

a) If event E cannot happen, then P(E) = 0.
b) If event E is certain to happen then P(E) = 1.
c) The probability that event E will occur is a number from 0 to 1: 0 ≤ P(E) ≤ 1.

For example, in a coin toss, the event that the coin lands on its edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.

Example 10 Let's assume that 2 cards are drawn from a 52-card deck. What is the probability that both of them are peaks?

Solution The number n of ways to draw 2 cards from a well-shuffled deck of 52 cards is 52 C 2 . Since 13 of the 52 cards are spades, the number of ways m to draw 2 spades is 13 C 2 . Then,
P(pulling 2 peaks) = m/n = 13 C 2 / 52 C 2 = 78/1326 = 1/17.

Example 11 Suppose 3 people are randomly selected from a group of 6 men and 4 women. What is the probability that 1 man and 2 women will be selected?

Solution The number of ways to select three people from a group of 10 people is 10 C 3. One man can be chosen in 6 C 1 ways, and 2 women can be chosen in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose 1 man and 2 women is 6 C 1. 4 C 2 . Then, the probability that 1 man and 2 women will be selected is
P = 6 C 1 . 4 C 2 / 10 C 3 = 3/10.

Example 12 Throwing dice. What is the probability of rolling a total of 8 on two dice?

Solution Each dice has 6 possible outcomes. The outcomes are doubled, meaning there are 6.6 or 36 possible ways in which the numbers on the two dice can appear. (It’s better if the cubes are different, say one is red and the other is blue - this will help visualize the result.)

The pairs of numbers that add up to 8 are shown in the figure below. There are 5 possible ways to obtain a sum equal to 8, hence the probability is 5/36.

Sometimes it is expressed as a percentage: P(A) 100% is the average percentage of the number of occurrences of an event A. Of course, it should be remembered that we are talking about some kind of mass operation, i.e. the conditions S test production - certain; if they change significantly, the probability of the event may change A: then will be the probability of the event A in a different mass operation, with different test conditions. In the future, we will assume, without specifying this each time, that we are talking about a certain mass operation; if the conditions under which the tests are carried out change, this will be specially noted.

37 Basic rules for finding the probability of an event.38,39

Combinatorics is a branch of mathematics that studies the question of how many

various combinations subject to certain conditions can be composed of

finite number of different elements.

Combinations that differ from each other in the composition of elements or their order

called compounds, there are three types of compounds.

Arrangements are connections made up of n-different elements according to

m-elements, which differ from each other either in the composition of elements or their

in order.

Permutations are compounds made up of the same n-elements,

which differ from each other only in their order of placement

Combinations are compounds made up of n-different elements of m-

elements that differ from each other by at least one element.

Combinations with repetitions are such compounds consisting of n-different

elements by m-elements differing from each other or at least by one

element or the fact that at least one element appears a different number of times

Sum Rule

If some object A can be selected from a set of objects M

ways, and object B in N ways, then choose either object A or object B

can be implemented in M+N ways.

Product rule

If object A can be selected from a set of objects in M ways, and then

such a choice, object B can be selected in N ways, then a pair of objects A and B

can be selected in A*B ways.

Basic concepts of probability theory

An event is any outcome of an experience; the following types of events are distinguished:

Random

Reliable

Impossible

The concept of a reliable and impossible event is used to quantify

assessing the possibility of the occurrence of a particular phenomenon, and with quantitative

The estimate is associated with probability.

The event is called incompatible in this experiment, if the appearance of one of

they are excluded by the appearance of another.

The event is called joint if the appearance of one of them does not exclude

the appearance of the others.

Several events form full group events if as a result of experience

At least one of them is sure to appear.

If two incompatible events form a complete group they are called

opposite

The event is called equally possible if the appearance of none of them

is objectively more possible than others.

The events are called unequally possible if the appearance of at least one of

them is more possible than others.

Cases are called incompatible equally possible and forming a complete

event group.

Calculating Probabilities

1. classic way

2. geometric

3. statistical

The first two methods are called direct counting methods

probability, and the classic one is based on counting the number of experiments

favorable for a given event among all its possible outcomes.

Basics of probability theory

The sum of events A i is the event C consisting in the occurrence of the event

A or events B or both of them together.

The sum of events A and B is the event C concluded in the execution of at least

one of the named events.

The product of several events is an event consisting of

joint implementation of all these events.

Probability multiplication theorem.

Event A is said to be dependent on event B if its probability varies

depending on whether event B occurred or not.

For independent events, the conditional and unconditional probabilities are the same.

The probability of occurrence of two dependent events is equal to the product of the probabilities

one of them by the probability of the other calculated under the condition that the first

the event took place.

P(A*B)=P(A)*P(B/A)=P(B)*P(B/A)

The probability of multiple events occurring is equal to the product of the probabilities

of these events and the probability of each subsequent event is calculated at

provided that all previous ones have taken place.

P(A 1;A 2.A n)=P(A 1)*P(A 2 /A 1)*.

*P(A n /A 1 ,A 2 .A n-1)

Theorem for adding probabilities of joint events

The probability of the sum of two joint events is equal to the sum of the probabilities of these

events without the probability of their co-occurrence.

P(A)+P(B)=P(A)+P(B)-P(A*B)

Probability of at least one event occurring

The probability of occurrence of event A consisting in the occurrence of at least one of

independent sets of events.A 1,A 2.A n

equal to the difference between one and the product of the probabilities of opposites

events A 1, A 2. A n

P(A)=1-q 1 *q 2 *.*q n

Total Probability Formula

Let event A appear together with one of the elements forming a complete group

pairwise incompatible events Н 1 ,Н 2 .Н n

called hypotheses, then the probability of event A is calculated as the sum

products of the probabilities of each hypothesis by the probability of event A for this

hypothesis

Bays formula

Let there be a complete group of pairwise inconsistent hypotheses H 1,H 2

Н n with known probabilities of occurrence. As a result of

experience, some event A appears, it is necessary to reestimate the probabilities of the hypotheses

provided that event A occurs

Repeating experiments

Several experiments are called independent if the probability of one or the other

of the outcomes of each of their experiments does not depend on what outcomes the others had

Theorem. If n independent experiments are performed in each of which

event A appears with the same probability p, and then the probability

the fact that event A will appear exactly m times is determined by the formula.

Bernoulli formula

Bernoulli's formula is used in cases where the number of experiments is small, and

the likelihood of occurrence is quite high.

If the number of trials n tends to 0, and the probability of occurrence of event A in each

from experiments p tends to 0, then to determine the probability of occurrence of event A

apply exactly m times Poisson's formula

If the number of experiments is large enough but not infinite, and the probability of occurrence

events A in each experiment does not tend to 0, local and integral

Laplace's theorem

Local Laplace theorem. The probability that in n independent

trials in each of which the probability of occurrence of event A is equal to p and

1>p>0, then this event occurs exactly m times approximately equal to

Laplace's integral theorem. The probability that in n independent

trials in each of which the probability of occurrence of event A is equal to p, and

1>p>0, then event A will occur at least m 1 times and no more than m

2 times is approximately equal

Random variables and laws of their distribution

Experience is any implementation of certain conditions and actions under

which the random phenomenon being studied is observed. Experiences can be characterized

qualitatively and quantitatively.

A random quantity is a quantity that, as a result of experiment, can take on

or another meaning, and it is not known in advance which one. Random

quantities are usually denoted (X,Y,Z), and the corresponding values (x,y,z)

Discrete are called random variables that take individual

values isolated from each other that can be overestimated.

Continuous quantities whose possible values are continuously filled

some range.

The law of distribution of a random variable is any relation

establishing a connection between possible values of random variables and

their corresponding probabilities.

Distribution row and polygon.

The simplest form of the law of distribution of a discrete quantity is the series

distributions.

The graphical interpretation of the distribution series is a polygon

distributions.

Distribution function of a random variable.

For continuous random variables, the following form of the distribution law is used:

as a distribution function.

The distribution function of the random variable X is called the function of the argument x,

that the random variable X takes any value less than x (X<х)

F(x)=P(X<х)

F(x) - sometimes called the cumulative distribution function or integral

law of distribution.

The distribution function has the following properties:

1. 0

2. if x 1 >x 2, then F(x 1)>F(x 2)

the function can be depicted as a graph. For a continuous value this is

there will be a curve varying from 0 to 1, and for a discrete value -

stepped figure with jumps.

Using the distribution function, you can easily find the probability of a hit

values for the area from α to β

Р(α<х<β) рассмотрим 3 события

B - α<Х<β

P(C)=P(A)+P(B)

Р(α<х<β)=Р(α)-Р(β)

Probability density distribution of a continuous random variable.

Probability distribution density of a continuous random variable X

is called a function f(x) equal to the first derivative of the distribution function

A graph of a distribution density is called a distribution curve.

Basic properties of the density distribution function:

Characteristics of the position of a random variable.

Fashion (Mo) of a random variable x is called its most probable

meaning. This definition strictly applies to discrete random variables.

For a continuous value fashion is called its value for which

The distribution density function has a maximum value.

Median (Me) a random variable is its value for

which random variable will be less than this value.

For a continuous random variable, the median is the abscissa of the point at which

The area under the curve is divided in half.

For a discrete random variable, the value of the median depends on whether it is even or

odd random value

n=2k+1, then Me=x k+1 (order average value)

If the value of random variables is even, i.e. n=2k, then

Mathematical expectation of a random variable.

The mathematical expectation of the random variable x ( M[x]) is called medium

weighted value of a random variable, with the weights being

the probability of occurrence of certain values.

For a discrete random variable

For continuous

From a mechanical point of view, mat. Expectation is the abscissa of the system's center of gravity

points located along the axis of the same name. Dimension mat. Expectations coincide with

the dimension of the random variable itself.

The mathematical expectation of a random variable is always greater than the smallest value

and less than the greatest.

Scattering characteristics.

Dispersion

Dispersion (D[x]) characterizes the dispersion or sparseness of random

values around its mathematical expectation.

For discrete

For continuous

The variance of a random variable is always positive

The variance dimension is equal to the squared difference of the random variable

Mean square (standard) deviation.

Some laws of distribution of random variables.

For discrete random variables - binomial distribution and distribution

Poisson

For continuous - uniform exponential, exponential and normal

distribution.

Binomial distribution.

Binomial is the law of distribution of the random variable X number

the occurrence of some event in n experiments if the probability p of the occurrence of the event

constant in every experience

The sum of the probabilities is represented by Newton's binomial

To determine numerical characteristics in the binomial distribution

substitute the probability which is determined by the Bernoulli formula.

With a binomial distribution, the variance is equal to mat. Expectation multiplied by

the probability of an event occurring in a particular experiment.

Poisson distribution

When you need to predict the expected queue and intelligently balance

number and productivity of service points and waiting time in queue.

The law of distribution of a discrete random variable X is called Poisson

the number of occurrence of some event in n-independent experiments if the probability

The fact that an event will appear exactly m times is determined by the formula.

n is the number of experiments performed

p-probability of occurrence of an event in each experiment

In queuing theory, the parameter of the Poisson distribution

determined by the formula

а=λt, where λ is the intensity of the message flow t-time

It should be noted that the Poisson distribution is limiting

case of binomial, when tests tend to infinity, and

the probability of an event occurring in each experiment tends to 0.

The Poisson distribution is the unit distribution for which

characteristics such as mat. Expectation and variance are the same and they are equal

parameter of this distribution law a.

Law of Uniform Density

The distribution of a continuous random variable X is called uniform

whose values lie on the segment and have a constant density

distribution

the area under the distribution curve is 1 and therefore c(b-a)=1

probability of random variable X falling on the interval from (α;β)

α=a, if α<а

β=в, if β>в

the main numerical characteristics of the density distribution law are calculated

according to general formulas and they are equal

Exponential (exponential distribution)

An exponential distribution is a distribution of a continuous random variable X which

is described by the following differential function

The exponential distribution for continuous random variables is

analogue of the Poisson distribution for discrete random variables and has

next view.

probability of random variable X falling on the interval (α;β)

It should be noted that the uptime is satisfied precisely

exponential law, and therefore this concept is often used in the concept

reliability.

Normal distribution law (Gauss's law)

The distribution of a random variable X is called normal if the density function

distribution

The resulting expression cannot be expressed through elementary functions, such

the so-called probability integral function for which tables have been compiled,

most often used as such a function

Often, according to the conditions of the problem, it is necessary to determine the probability of hitting a random

X value per area symmetrical to the mathematical expectation.

The Three Sigma Rule is often used to confirm or

rejecting the hypothesis of normal distribution of a random variable.

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 giving good luck to 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 happen. 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 certain 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.

Mom washed the frame

At the end of the long summer holidays, it’s time to slowly return to higher mathematics and solemnly open the empty Verdov file to begin creating a new section - . I admit, the first lines are not easy, but the first step is half the way, so I suggest everyone carefully study the introductory article, after which mastering the topic will be 2 times easier! I'm not exaggerating at all. …On the eve of the next September 1st, I remember first grade and the primer…. Letters form syllables, syllables form words, words form short sentences - Mom washed the frame. Mastering turver and math statistics is as easy as learning to read! However, for this you need to know key terms, concepts and designations, as well as some specific rules, which are the subject of this lesson.

But first, please accept my congratulations on the beginning (continuation, completion, mark as appropriate) of the school year and accept the gift. The best gift is a book, and for independent work I recommend the following literature:

1) Gmurman V.E. Theory of Probability and Mathematical Statistics

A legendary textbook that has gone through more than ten reprints. It is distinguished by its intelligibility and extremely simple presentation of the material, and the first chapters are completely accessible, I think, already for students in grades 6-7.

2) Gmurman V.E. Guide to solving problems in probability theory and mathematical statistics

A solution book by the same Vladimir Efimovich with detailed examples and problems.

NECESSARILY download both books from the Internet or get their paper originals! The version from the 60s and 70s will also work, which is even better for dummies. Although the phrase “probability theory for dummies” sounds rather ridiculous, since almost everything is limited to elementary arithmetic operations. They skip, however, in places derivatives And integrals, but this is only in places.

I will try to achieve the same clarity of presentation, but I must warn that my course is aimed at problem solving and theoretical calculations are kept to a minimum. Thus, if you need a detailed theory, proofs of theorems (yes, theorems!), please refer to the textbook.

For those who want learn to solve problems in a matter of days, created crash course in pdf format (based on site materials). Well, right now, without putting things off for a long time, we are starting to study terver and matstat - follow me!

That's enough for a start =)

As you read the articles, it is useful to become familiar (at least briefly) with additional tasks of the types considered. On the page Ready-made solutions for higher mathematics The corresponding pdfs with examples of solutions are posted. Significant assistance will also be provided IDZ 18.1-18.2 Ryabushko(simpler) and solved IDZ according to Chudesenko’s collection(more difficult).

1) Amount two events and the event is called which is that it will happen or event or event or both events at the same time. In the event that events incompatible, the last option disappears, that is, it may occur or event or event .

The rule also applies to a larger number of terms, for example, the event is what will happen at least one from events , A if events are incompatible – then one thing and only one thing event from this amount: or event , or event , or event , or event , or event .

There are plenty of examples:

Events (when throwing a dice, 5 points will not appear) is what will appear or 1, or 2, or 3, or 4, or 6 points.

Event (will drop no more two points) is that 1 will appear or 2points.

Event (there will be an even number of points) is what appears or 2 or 4 or 6 points.

The event is that a red card (heart) will be drawn from the deck or tambourine), and the event – that the “picture” will be extracted (jack or lady or king or ace).

A little more interesting is the case with joint events:

The event is that a club will be drawn from the deck or seven or seven of clubs According to the definition given above, at least something- or any club or any seven or their “intersection” - seven of clubs. It is easy to calculate that this event corresponds to 12 elementary outcomes (9 club cards + 3 remaining sevens).

The event is that tomorrow at 12.00 will come AT LEAST ONE of the summable joint events, namely:

– or there will be only rain / only thunderstorm / only sun;
– or only some pair of events will occur (rain + thunderstorm / rain + sun / thunderstorm + sun);
– or all three events will appear simultaneously.

That is, the event includes 7 possible outcomes.

The second pillar of the algebra of events:

2) The work two events and call an event which consists in the joint occurrence of these events, in other words, multiplication means that under some circumstances there will be And event , And event . A similar statement is true for a larger number of events, for example, a work implies that under certain conditions it will happen And event , And event , And event , …, And event .

Consider a test in which two coins are tossed and the following events:

– heads will appear on the 1st coin;
– the 1st coin will land heads;
– heads will appear on the 2nd coin;
– the 2nd coin will land heads.

Then:
And on the 2nd) heads will appear;
– the event is that on both coins (on the 1st And on the 2nd) it will be heads;
– the event is that the 1st coin will land heads And the 2nd coin is tails;
– the event is that the 1st coin will land heads And on the 2nd coin there is an eagle.

It is easy to see that events incompatible (because, for example, it cannot be 2 heads and 2 tails at the same time) and form full group (since taken into account All possible outcomes of tossing two coins). Let's summarize these events: . How to interpret this entry? Very simple - multiplication means a logical connective AND, and addition – OR. Thus, the amount is easy to read in understandable human language: “two heads will appear or two heads or the 1st coin will land heads And on the 2nd tails or the 1st coin will land heads And on the 2nd coin there is an eagle"

This was an example when in one test several objects are involved, in this case two coins. Another common scheme in practical problems is retesting , when, for example, the same die is rolled 3 times in a row. As a demonstration, consider the following events:

– in the 1st throw you will get 4 points;
– in the 2nd throw you will get 5 points;
– in the 3rd throw you will get 6 points.

Then the event is that in the 1st throw you will get 4 points And in the 2nd throw you will get 5 points And on the 3rd roll you will get 6 points. Obviously, in the case of a cube there will be significantly more combinations (outcomes) than if we were tossing a coin.

...I understand that perhaps the examples being analyzed are not very interesting, but these are things that are often encountered in problems and there is no escape from them. In addition to a coin, a cube and a deck of cards, urns with multi-colored balls, several anonymous people shooting at a target, and a tireless worker who is constantly grinding out some details await you =)

Probability of event

Probability of event is the central concept of probability theory. ...A killer logical thing, but we had to start somewhere =) There are several approaches to its definition:

;
Geometric definition of probability ;
Statistical definition of probability .

In this article I will focus on the classical definition of probability, which is most widely used in educational tasks.

Designations. The probability of a certain event is indicated by a capital Latin letter, and the event itself is taken in brackets, acting as a kind of argument. For example:

Also, the small letter is widely used to denote probability. In particular, you can abandon the cumbersome designations of events and their probabilities in favor of the following style::

– the probability that a coin toss will result in heads;
– the probability that a dice roll will result in 5 points;
– the probability that a card of the club suit will be drawn from the deck.

This option is popular when solving practical problems, since it allows you to significantly reduce the recording of the solution. As in the first case, it is convenient to use “talking” subscripts/superscripts here.

Everyone has long guessed the numbers that I just wrote down above, and now we will find out how they turned out:

Classic definition of probability:

The probability of an event occurring in a certain test is called the ratio , where:

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

- quantity elementary outcomes, favorable event.

When tossing a coin, either heads or tails can fall out - these events form full group, thus, the total number of outcomes; at the same time, each of them elementary And equally possible. The event is favored by the outcome (heads). According to the classical definition of probability: .

Similarly, as a result of throwing a die, elementary equally possible outcomes may appear, forming a complete group, and the event is favored by a single outcome (rolling a five). That's why: THIS IS NOT ACCEPTED TO DO (although it is not forbidden to estimate percentages in your head).

It is customary to use fractions of a unit, and, obviously, the probability can vary within . Moreover, if , then the event is impossible, If - reliable, and if , then we are talking about random event.

! If, while solving any problem, you get some other probability value, look for the error!

In the classical approach to determining probability, extreme values (zero and one) are obtained through exactly the same reasoning. Let 1 ball be drawn at random from a certain urn containing 10 red balls. Consider the following events:

in a single trial a low-possibility event will not occur.

This is why you will not hit the jackpot in the lottery if the probability of this event is, say, 0.00000001. Yes, yes, it’s you – with the only ticket in a particular circulation. However, a larger number of tickets and a larger number of drawings will not help you much. ...When I tell others about this, I almost always hear in response: “but someone wins.” Okay, then let's do the following experiment: please buy a ticket for any lottery today or tomorrow (don't delay!). And if you win... well, at least more than 10 kilorubles, be sure to sign up - I will explain why this happened. For a percentage, of course =) =)

But there is no need to be sad, because there is an opposite principle: if the probability of some event is very close to one, then in a single trial it will almost certain will happen. Therefore, before jumping with a parachute, there is no need to be afraid, on the contrary, smile! After all, completely unthinkable and fantastic circumstances must arise for both parachutes to fail.

Although all this is lyricism, since depending on the content of the event, the first principle may turn out to be cheerful, and the second – sad; or even both are parallel.

Perhaps that's enough for now, in class Classical probability problems we will get the most out of the formula. In the final part of this article, we will consider one important theorem:

The sum of the probabilities of events that form a complete group is equal to one. Roughly speaking, if events form a complete group, then with 100% probability one of them will happen. In the simplest case, a complete group is formed by opposite events, for example:

– as a result of a coin toss, heads will appear;
– the result of a coin toss will be heads.

According to the theorem:

It is absolutely clear that these events are equally possible and their probabilities are the same .

Due to the equality of probabilities, equally possible events are often called equally probable . And here is a tongue twister for determining the degree of intoxication =)

Example with a cube: events are opposite, therefore .

The theorem under consideration is convenient in that it allows you to quickly find the probability of the opposite event. So, if the probability that a five is rolled is known, it is easy to calculate the probability that it is not rolled:

This is much simpler than summing up the probabilities of five elementary outcomes. For elementary outcomes, by the way, this theorem is also true:
. For example, if is the probability that the shooter will hit the target, then is the probability that he will miss.

! In probability theory, it is undesirable to use letters for any other purposes.

In honor of Knowledge Day, I will not assign homework =), but it is very important that you can answer the following questions:

– What types of events exist?
– What is chance and equal possibility of an event?
– How do you understand the terms compatibility/incompatibility of events?
– What is a complete group of events, opposite events?
– What does addition and multiplication of events mean?
– What is the essence of the classical definition of probability?
– Why is the theorem for adding the probabilities of events that form a complete group useful?

No, you don’t need to cram anything, these are just the basics of probability theory - a kind of primer that will quickly fit into your head. And for this to happen as soon as possible, I suggest you familiarize yourself with the lessons