Taking probability. Probability theory. Probability of an event, random events (probability theory). Independent and incompatible events in probability theory. Examples of solutions to problems on classical probability

then we say that the random variable ξ It has probability distribution density f(x).

Definition. Let . For a continuous distribution function F theoretical α-quantile is called the solution of the equation.

This solution may not be the only one.

Level quantile ½ called theoretical median , level quantiles ¼ And ¾ -lower and upper quartiles respectively.

In actuarial applications, an important role is played by Chebyshev's inequality:

for any

Mathematical expectation symbol.

It reads like this: the probability that modulus is greater than less than or equal to the expectation of modulus divided by .

Lifetime as a random variable

The uncertainty of the moment of death is a major risk factor in life insurance.

Nothing definite can be said about the moment of death of an individual. However, if we are dealing with a large homogeneous group of people and are not interested in the fate of individual people from this group, then we are within the framework of probability theory as a science of mass random phenomena with the frequency stability property.

Respectively, we can talk about life expectancy as a random variable T.

survival function

In probability theory, they describe the stochastic nature of any random variable T distribution function F(x), which is defined as the probability that the random variable T less than number x:

.

In actuarial mathematics, it is pleasant to work not with a distribution function, but with an additional distribution function . In terms of longevity, it is the probability that a person will live to the age x years.

called survival function(survival function):

The survival function has the following properties:

In life tables, it is usually assumed that there is some age limit (limiting age) (as a rule, years) and, accordingly, at x>.

When describing mortality by analytical laws, it is usually assumed that the life time is unlimited, however, the type and parameters of the laws are selected so that the probability of life over a certain age is negligible.

The survival function has a simple statistical meaning.

Let's say that we are observing a group of newborns (usually ) whom we observe and can record the moments of their death.

Let us denote the number of living representatives of this group in age through . Then:

.

Symbol E here and below is used to denote the mathematical expectation.

So, the survival function is equal to the average proportion of those who survived to age from a certain fixed group of newborns.

In actuarial mathematics, one often works not with a survival function, but with a value just introduced (having fixed the initial group size).

The survival function can be reconstructed from the density:

Life span characteristics

From a practical point of view, the following characteristics are important:

1 . Average lifetime

,
2 . Dispersion lifetime

,
Where
,

Presented to date in the open bank of USE problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is a classical definition of probability.

The easiest way to understand the formula is with examples.
Example 1 There are 9 red balls and 3 blue ones in the basket. The balls differ only in color. At random (without looking) we get one of them. 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 the ball) that can have a different result - an outcome. It should be noted that the result can be viewed in different ways. "We pulled out a ball" is also a result. "We pulled out the blue ball" is the result. "We drew this particular ball out of all possible balls" - this least generalized view of the result is called the elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.

Solution. Now we calculate the probability of choosing a blue ball.
Event A: "the chosen ball turned out to be blue"
Total number of all possible outcomes: 9+3=12 (number of all balls 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

Let us calculate for the same problem the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. The number of favorable outcomes: 9. The desired probability: 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 mathematical and conversational assessment is done by multiplying (or dividing) by 100%.
So,
In this case, the probability is zero for events that cannot happen - improbable. 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 counted according to the formula)
Probability 1 has events that will absolutely definitely happen, without options. For example, the probability that "the chosen ball will be either red or blue" is for our problem. (Number of favorable outcomes: 12, P(A)=12/12=1)

We've looked at a classic example that illustrates the definition of probability. All similar USE problems in probability theory are solved using this formula.
Instead of red and blue balls, there can be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on a 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 USE probability theory, where you need to calculate the probability of an event occurring on a certain day. ( , ) As in the previous tasks, you need to determine what is an elementary outcome, and then apply the same formula.

Example 2 The conference lasts three days. On the first and second days, 15 speakers each, on the third day - 20. What is the probability that the report of Professor M. will fall on the third day, if the order of the reports is determined by lottery?

What is the elementary outcome here? - Assigning a professor's report to one of all possible serial numbers for a speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report can receive one of 50 numbers. This means that 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, the probability P(A)= 20/50=2/5=4/10=0.4
Answer: 0.4

The drawing of lots here is the establishment of a random correspondence between people and ordered places. In Example 2, matching was considered in terms of which of the places a particular person could take. You can approach the same situation from the other side: which of the people with what probability could get to a particular place (prototypes , , , ):

Example 3 5 Germans, 8 Frenchmen and 3 Estonians participate in the draw. 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 to a given place by lot. 5+8+3=16 people.
Favorable outcomes - the French. 8 people.
Desired probability: 8/16=1/2=0.5
Answer: 0.5

The prototype is slightly different. There are tasks about coins () and dice () that are somewhat more creative. Solutions to these problems can be found on the prototype pages.

Here are some examples of tossing a coin or a die.

Example 4 When we toss a coin, what is the probability of getting tails?
Outcomes 2 - heads or tails. (it is believed that the coin never falls on the edge) Favorable outcome - tails, 1.
Probability 1/2=0.5
Answer: 0.5.

Example 5 What if we flip a coin twice? What is the probability that it will come up heads both times?
The main thing is to determine which 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 tails
2) PO - first time tails, second time heads
3) OP - the first time heads, the second time tails
4) OO - heads up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one is favorable, 1.
Probability: 1/4=0.25
Answer: 0.25

What is the probability that two tosses of a coin will land on 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 come in handy.
If at one toss of a coin we have 2 possible outcomes, then for two tosses of results there 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 throws of possible outcomes there will be 2·2·...·2=2 N .

So, you can find the probability of getting 5 tails out of 5 coin tosses.
The total number of elementary outcomes: 2 5 =32.
Favorable outcomes: 1. (RRRRRR - all 5 times tails)
Probability: 1/32=0.03125

The same is true for the 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 a dice. What is the probability of getting an even number?

Total outcomes: 6, according to the number of faces.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6=0.5

Example 7 Throw two dice. What is the probability that the total rolls 10? (round to hundredths)

There are 6 possible outcomes for one die. Hence, for two, according to the above rule, 6·6=36.
What outcomes will be favorable for a total of 10 to fall out?
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. So, for cubes, options are possible:
(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)
In total, 3 options. Desired probability: 3/36=1/12=0.08
Answer: 0.08

Other types of B6 problems will be discussed in one of the following "How to Solve" articles.

probability is a number from 0 to 1 that reflects the chances that a random event will occur, where 0 is the complete absence of the probability of the occurrence of the event, and 1 means that the event in question will definitely occur.

The probability of an event E is a number between and 1.
The sum of the probabilities of mutually exclusive events is 1.

empirical probability- probability, which is calculated as the relative frequency of the event in the past, extracted from the analysis of historical data.

The probability of very rare events cannot be calculated empirically.

subjective probability- the probability based on a personal subjective assessment of the event, regardless of historical data. Investors who make decisions to buy and sell stocks often act on the basis of subjective probability.

prior probability -

Chance 1 out of… (odds) that an event will occur through the concept of probability. The chance of an event occurring is expressed in terms of probability as follows: P/(1-P).

For example, if the probability of an event is 0.5, then the chance of an event is 1 out of 2, since 0.5/(1-0.5).

The chance that the event will not occur is calculated by the formula (1-P)/P

Inconsistent Probability- for example, in the price of shares of company A, 85% of the possible event E is taken into account, and in the price of shares of company B, only 50%. This is called the mismatched probability. According to the Dutch Betting Theorem, mismatched probability creates opportunities for profit.

Unconditional Probability is the answer to the question "What is the probability that the event will occur?"

Conditional Probability is the answer to the question: "What is the probability of event A if event B happened." The conditional probability is denoted as P(A|B).

Joint Probability is the probability that events A and B will happen at the same time. Designated as P(AB).

P(A|B) = P(AB)/P(B) (1)

P(AB) = P(A|B)*P(B)

Probability summation rule:

The probability that either event A or event B will happen is

P(A or B) = P(A) + P(B) - P(AB) (2)

If events A and B are mutually exclusive, then

P(A or B) = P(A) + P(B)

Independent events- events A and B are independent if

P(A|B) = P(A), P(B|A) = P(B)

That is, it is a sequence of outcomes where the probability value is constant from one event to the next.
A coin toss is an example of such an event - the result of each next toss does not depend on the result of the previous one.

Dependent events These are events in which the probability of one occurring depends on the probability of the other occurring.

Rule for multiplying the probabilities of independent events:
If events A and B are independent, then

P(AB) = P(A) * P(B) (3)

Total Probability Rule:

P(A) = P(AS) + P(AS") = P(A|S")P(S) + P(A|S")P(S") (4)

S and S" are mutually exclusive events

expected value random variable is the average of the possible outcomes of the random variable. For the event X, the expectation is denoted as E(X).

Suppose we have 5 values ​​​​of mutually exclusive events with a certain probability (for example, the company's income amounted to such and such an amount with such a probability). The expectation is the sum of all outcomes multiplied by their probability:

The variance of a random variable is the expected value of square deviations of a random variable from its expected value:

s 2 = E( 2 ) (6)

Conditional expected value - the expectation of a random variable X, provided that the event S has already occurred.

From a practical point of view, event probability is the ratio of the number of those observations in which the event in question occurred to the total number of observations. Such an interpretation is admissible in the case of a sufficiently large number of observations or experiments. For example, if about half of the people you meet on the street are women, then you can say that the probability that the person you meet on the street is a woman is 1/2. In other words, the frequency of its occurrence in a long series of independent repetitions of a random experiment can serve as an estimate of the probability of an event.

Probability in mathematics

In the modern mathematical approach, the classical (that is, not quantum) probability is given by Kolmogorov's axiomatics. Probability is a measure P, which is set on the set X, called the probability space. This measure must have the following properties:

It follows from these conditions that the probability measure P also has the property additivity: if sets A 1 and A 2 do not intersect, then . To prove it, you need to put everything A 3 , A 4 , … equal to the empty set and apply the property of countable additivity.

The probability measure may not be defined for all subsets of the set X. It suffices to define it on the sigma-algebra consisting of some subsets of the set X. In this case, random events are defined as measurable subsets of the space X, that is, as elements of the sigma algebra.

Probability sense

When we find that the reasons for some possible fact to actually occur outweigh the opposite reasons, we consider this fact probable, otherwise - incredible. This predominance of positive bases over negative ones, and vice versa, can represent an indefinite set of degrees, as a result of which probability(And improbability) It happens more or less .

Complicated single facts do not allow an exact calculation of their degrees of probability, but even here it is important to establish some large subdivisions. So, for example, in the field of law, when a personal fact subject to trial is established on the basis of witness testimony, it always remains, strictly speaking, only probable, and it is necessary to know how significant this probability is; in Roman law, a quadruple division was accepted here: probatio plena(where the probability practically turns into authenticity), Further - probatio minus plena, then - probatio semiplena major and finally probatio semiplena minor .

In addition to the question of the probability of the case, there may arise, both in the field of law and in the field of morality (with a certain ethical point of view), the question of how likely it is that a given particular fact constitutes a violation of the general law. This question, which serves as the main motive in the religious jurisprudence of the Talmud, gave rise in Roman Catholic moral theology (especially from the end of the 16th century) to very complex systematic constructions and an enormous literature, dogmatic and polemical (see Probabilism).

The concept of probability admits of a definite numerical expression in its application only to such facts which are part of certain homogeneous series. So (in the simplest example), when someone throws a coin a hundred times in a row, we find here one common or large series (the sum of all falls of a coin), which is composed of two private or smaller, in this case numerically equal, series (falls " eagle" and falling "tails"); The probability that this time the coin will fall tails, that is, that this new member of the general series will belong to this of the two smaller series, is equal to a fraction expressing the numerical ratio between this small series and the large one, namely 1/2, that is, the same probability belongs to one or the other of the two private series. In less simple examples, the conclusion cannot be drawn directly from the data of the problem itself, but requires prior induction. So, for example, it is asked: what is the probability for a given newborn to live up to 80 years? Here there must be a general or large series of a known number of people born in similar conditions and dying at different ages (this number must be large enough to eliminate random deviations, and small enough to preserve the homogeneity of the series, because for a person, born, for example, in St. Petersburg in a well-to-do cultural family, the entire million-strong population of the city, a significant part of which consists of people from various groups that can die prematurely - soldiers, journalists, workers in dangerous professions - represents a group too heterogeneous for a real definition of probability) ; let this general series consist of ten thousand human lives; it includes smaller rows representing the number of those who live to this or that age; one of these smaller rows represents the number of those living to 80 years of age. But it is impossible to determine the size of this smaller series (as well as all others). a priori; this is done in a purely inductive way, through statistics. Suppose statistical studies have established that out of 10,000 Petersburgers of the middle class, only 45 survive to the age of 80; thus, this smaller row is related to the larger one as 45 to 10,000, and the probability for a given person to belong to this smaller row, that is, to live to 80 years old, is expressed as a fraction of 0.0045. The study of probability from a mathematical point of view constitutes a special discipline, the theory of probability.

see also

Notes

Literature

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See what "Probability" is in other dictionaries:

General scientific and philosophical. a category denoting the quantitative degree of the possibility of the appearance of mass random events under fixed observation conditions, characterizing the stability of their relative frequencies. In logic, the semantic degree ... ... Philosophical Encyclopedia

PROBABILITY, a number in the range from zero to one, inclusive, representing the possibility of this event happening. The probability of an event is defined as the ratio of the number of chances that an event can occur to the total number of possible ... ... Scientific and technical encyclopedic dictionary

In all likelihood .. Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M.: Russian dictionaries, 1999. probability, possibility, probability, chance, objective possibility, maza, admissibility, risk. Ant. impossibility... ... Synonym dictionary

probability- A measure that an event can occur. Note The mathematical definition of probability is "a real number between 0 and 1 relating to a random event." The number may reflect the relative frequency in a series of observations ... ... Technical Translator's Handbook

Probability- "a mathematical, numerical characteristic of the degree of possibility of the occurrence of any event in certain specific conditions that can be repeated an unlimited number of times." Based on this classic… … Economic and Mathematical Dictionary

- (probability) The possibility of an event or a certain result occurring. It can be represented as a scale with divisions from 0 to 1. If the probability of an event is zero, its occurrence is impossible. With a probability equal to 1, the onset of ... Glossary of business terms

In the USE assignments in mathematics, there are also more complex probability tasks (than we considered in Part 1), where you have to apply the rule of addition, multiplication of probabilities, and distinguish between joint and incompatible events.

So, theory.

Joint and non-joint events

Events are said to be incompatible if the occurrence of one of them excludes the occurrence of the others. That is, only one particular event can occur, or another.

For example, by throwing a die, you can distinguish between events such as an even number of points and an odd number of points. These events are incompatible.

Events are called joint if the occurrence of one of them does not exclude the occurrence of the other.

For example, when throwing a die, you can distinguish between events such as the occurrence of an odd number of points and the loss of a number of points that is a multiple of three. When three is rolled, both events are realized.

Sum of events

The sum (or union) of several events is an event consisting in the occurrence of at least one of these events.

Wherein the sum of two disjoint events is the sum of the probabilities of these events:

For example, the probability of getting 5 or 6 points on a dice in one throw will be, because both events (drop 5, drop 6) are incompatible and the probability of one or the second event is calculated as follows:

The probability the sum of two joint events is equal to the sum of the probabilities of these events without taking into account their joint occurrence:

For example, in a shopping mall, two identical vending machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Let's find the probability that by the end of the day coffee will end in at least one of the machines (that is, either in one, or in the other, or in both at once).

The probability of the first event "coffee will end in the first machine" as well as the probability of the second event "coffee will end in the second machine" by the condition is equal to 0.3. Events are collaborative.

The probability of the joint realization of the first two events is equal to 0.12 according to the condition.

This means that the probability that by the end of the day the coffee will run out in at least one of the machines is

Dependent and independent events

Two random events A and B are called independent if the occurrence of one of them does not change the probability of the other occurring. Otherwise, events A and B are called dependent.

For example, when two dice are rolled at the same time, one of them, say 1, and the other 5, are independent events.

Product of probabilities

A product (or intersection) of several events is an event consisting in the joint occurrence of all these events.

If there are two independent events A and B with probabilities P(A) and P(B), respectively, then the probability of the realization of events A and B is simultaneously equal to the product of the probabilities:

For example, we are interested in the loss of a six on a dice twice in a row. Both events are independent and the probability of each of them occurring separately is . The probability that both of these events will occur will be calculated using the above formula: .

See a selection of tasks for working out the topic.