The Bayes formula looks like this.

What is the significance of the fact that they both received a “good” ticket? 117. Statistics on loan requests from the bank are as follows: 10% - holding.

organs, 20% - other banks, others - individuals.

The rate of non-repayment of loans is consistently 0.01, 0.05 and 0.2. Repair and self-development How many loans don't turn over? 118. 32 letters of the Russian alphabet are written on cut-out cards. Five cards are taken out one by one and placed on the table in the order they appear.

Find out the certainty of the word “end”. calculated according to the formula

119 Statistics on loan requests from the bank are as follows: 10% – holding.

organs, 20% - other banks, others - individuals. Repair and self-development The rate of non-repayment of loans is consistently 0.01, 0.05 and 0.2. How many loans don't turn over? 120. It is possible for a seller’s current turnover of frozen goods to exceed 2000 rubles. Repair and self-development become 80% in clear weather, 50% in cloudy weather and 10% in rainy weather.

How can you transfer the turnover to 2000 rubles? Repair and self-development The probability of clear weather is 20%, and the probability of cloudy weather and rainfall is 40% each. Repair and self-development Meta robots:

formulate skills for solving problems from the theory of probability using the additional formula of full probability and the Bayes formula.

This formula is called the Bayes formula (in the name of the English mathematician known as Viviv; published in 1764). Repair and self-development.

Bayes' formula allows us to re-evaluate the validity of hypotheses after we know the result of testing, which results in a hypothesis Zavdannya 1.

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

The part is inspected by one inspector;

It detects a defect with an authenticity of 0.97, and if the defect is not detected, the part is passed through the finished product. In addition, the inspector can gently reject the part if there is no defect;

It detects a defect with an authenticity of 0.97, and if the defect is not detected, the part is passed through the finished product. Its validity is 0.01.

Know the certainty of the steps: A – the part will be rejected; B – the part will be rejected, otherwise it will be taken care of; C – the part will be missed from the finished product with a defect.

Decision

Significant hypotheses: 0,01; N 0,97.

= (A standard part is needed for inspection);

= (We hope to control a non-standard part).

Podia

A =

(The part will be rejected). In our minds we know the importance of

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

Pfi(A) = Behind the formula of total strength comes

The probability that the part will be rejected by Milkovo is ancient We know how likely it is that a part will be missed from a finished product with a defect:

Subject:

Zavdannya 2.

The virus is checked for standardness by one of three merchants.

A = The probability of what you choose to spend to the first merchant is 0.25, to the second – 0.26, and to the third – 0.49.

The probability that the virus will be recognized by the standard first merchant is 0.95, by others – 0.98, and by the third – 0.97. Find out whether the standard virus has been verified by another controller.

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

Significantly: L. =

New products from three enterprises arrived at the store.(for verification, go to the /th merchant);

Zavdannya 2.

/ = 1, 2, 3;

B =

A = 0,75.

(virib will be recognized as standard). A device is being tested that consists of two nodes, the reliability of which is as high as 08 and 09.

• Vuzley suggests one kind of another.
• Prilad vidmoviv.
• Know the validity of these hypotheses:

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

a) faulty lishe first school; b) the other university is faulty; = 1,2;

c) incorrect offense vuzli.

Repair and self-development D = (7th school not in good order);

i D – vertical beds; 2) = 0,9.

= (When tried, I will adjust it).

Know the certainty of the steps: A – the part will be rejected; Repair and self-development The mind is obsessed with obsessiveness: P(D) = 0.8;

R(L

For the variety of properties of bedridden diseases

A =

modern sums of independent works Vikorist's theorem on the folding of the imaginary properties of non-crazy processes and the multiplication theorem of the multiplication of the non-independent ones, can be removed

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

Now we know the validity of the hypotheses: b) the other university is faulty; = 1, 2, 3;

New products from three enterprises arrived at the store. Zavdannya 5.

At the factory, bolts are prepared on three benches, which vibrate approximately 25%, 30% and 45% of the bolts’ weight.

Verstat products have a rate of 4%, 3% and 2%.

/ = 1, 2, 3;

A = 0,028.

How likely is it that a bolt, if taken from the product that was found, would turn out to be defective? 4 = (randomly take the bolt and prepare on the /th workbench);

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

= (Assuming the bolt turns out to be defective). In mind, behind the formula of classical reliability, we know the validity of hypotheses: 1, 2, 3;

New products from three enterprises arrived at the store. Also, behind the formula of classical intelligence we can find intellectual intelligence:

Zavdannya 6.

The electronic circuit should be used in one of three batches with a quality of 0.25;

/ = 1, 2, 3;

A = 0,225.

0.5 and 0.25. The reliability of the fact that the scheme applies to the guarantee term of service for each party is likely to become 0.1;

The plant produces the first type of part, the leather part has a defect with a level of 0.05.

RN(A) =

0.2 and 0.4. b) the other university is faulty; = 1,2;

Repair and self-development Find out whether the scheme you have taken is based on the warranty term of the service.

4 = (scheme from the ith batch was taken at random);

Know the certainty of the steps: A – the part will be rejected; Repair and self-development i = = (taken at random, the scheme applies to the warranty term of the service). Behind the mind's eye, there is a certainty of hypotheses: Repair and self-development Also in terms of mental intelligence: Repair and self-development 2 .

Zavdannya 7.

The device is used to mix two blocks; skin care is necessary for the operation of the device.

A =

Table 1 Bayes' formula allows us to re-evaluate the validity of hypotheses after we know the result of testing, which results in a hypothesis In the warehouse of the television studio there are 70% of kinescopes produced by plant No. 1;

(The part will be rejected). Other kinescopes are manufactured by plant No. 2. The reliability is that the kinescope does not go out of use during the warranty period of service, which is higher than 0.8 for kinescopes from plant No. 1 and 0.7 - for kinescopes in plant No. 2. The kinescope is subject to the warranty term of service.

Find out the reliability of what is produced by plant No. 2. The folding machine contains parts from three automatic machines.

(virib will be recognized as standard). Apparently, the first machine gives 0.3%, the second - 0.2%, the third - 0.4%.

modern sums of independent works Find out the possibility of recovering folded defective parts, if 1000 parts were found from the 1st machine, 2000 from the 2nd machine, 2500 parts from the 3rd machine.

The probability that the virus will be recognized by the standard first merchant is 0.95, by others – 0.98, and by the third – 0.97. Zavdannya 3.

0.5 and 0.25. However, the new parts are being prepared on two benches.

The reliability of the fact that the part was prepared on the first versat, being standard, is 0.8, and on the other - 0.9. The productivity of the other versat is three times greater than the productivity of the first. Know the reliability of what will be a standard part, taken from the conveyor, which will receive parts from both benches. The company's carrier plans to quickly employ the services of two of three transport companies. Know the reliability of what will be a standard part, taken from the conveyor, which will receive parts from both benches. The probability of untimely delivery to vantage for the first, second and third companies is equal to 0.05; 94. behind a cosmic object, the object appears with realism 0.1 and 0.07. Know the reliability of what will be a standard part, taken from the conveyor, which will receive parts from both benches. Having combined this data with data about the safety of important transportation, the carer made a decision about the fairness of the choice and decided to work for the foal. Know the truth of the fact that the first round is laid down, it’s also white. occurs in 1% of cases. Find out the severity of the defect R

for all products. Zavdannya 9.

The company has 10 new cars and 5 old cars that were previously under repair. The reliability of help robots for a new car is 0.94, for an old car – 0.91.

Find out the certainty that the car will be able to operate properly. Zavdannya 10.

Two sensors send signals into the back channel of the connection, and the first one sends twice as many signals as the other. The probability of rejecting the signal from the first sensor is 0.01, while the other is 0.03.

How can you cancel the signal from the front channel? Zavdannya 11.

Є five batches of viruses: three batches of 8 pieces, including 6 standard and 2 non-standard, and two batches of 10 pieces, including 7 standard and 3 non-standard. It’s natural to rob one of the lots, and take the part from that lot.

It is significant that the part taken will be standard. Zavdannya 12.

• The warehouse takes on average 50% of the parts from the first plant, 30% from another plant, and 20% from the third plant.
• b) The team did not win any points.

Know the truth that Sidorov spoke. Zavdannya 16.

The white screen has 12 hearts and 6 blue coins. The black one has 15 red coins and 10 blue ones.

Throw a dice. If the number of points is a multiple of 3, then it is tempting to take the point from the white box.

If you don’t have another big point, then it’s tempting to take a point from the black screen. What is the probability of the appearance of a red bag?

Zavdannya 17. Two boxes have radio tubes.

another has 10 lamps, їх 1 is non-standard.
A lamp was taken from the first box and transferred to another.

Know that the lamp that was pulled from another box will be non-standard.
a priori certainties Two hypotheses are possible: H 1 - device for collecting from high-acidity parts;

H 2 - device for collecting from parts of primary capacity.
a priori certainties The validity of these hypotheses is complete: P(H 1) = 0.4, P(H 2) = 0.6.
As a result, the method A was carefully guarded – the method was used without any warning at the hour t.
The level of validity of the price for the hypotheses H 1 and H 2 are equal: P(A|H 1) = 0.95;
P(A|H 2) = 0.7.
Using formula (12), we find the validity of hypothesis H 1 after evidence:
Butt No. 2.

Two shooters independently shoot at the same target, shooting the skin one shot at a time.
The likelihood of hitting the mark for the first shooter is 0.8, for the other 0.4.

Assuming that it is impossible for two arrows to hit the same point, find the confidence of those who hit the first arrow at the target.
a priori certainties Use shot A - after shooting, one hole was detected in the target.
To begin with, there are some possible hypotheses:
H 1 - neither the first nor the other arrow is lost, the validity of this hypothesis: P(H 1) = 0.2 · 0.6 = 0.12.
H 2 - burn off the arrows, P(H 2) = 0.8 · 0.4 = 0.32.
H 3 - the first shooter is included, but the other is not included, P(H 3) = 0.8 · 0.6 = 0.48.
H 4 - the first shooter is not received, but the other is lost, P (H 4) = 0.2 · 0.4 = 0.08.

The rationale for these hypotheses is equal:

We know the virality

Butt No. 4.

The reliability of the fact that at the time of operation of the system, which consists of three elements, elements numbered 1, 2 and 3 are shown as 3: 2: 5. The probability of identifying the types of these elements is Totally 0.95;

0.9 and 0.6.
b) In the minds of this department, a robotic system was revealed to Vidmov.
Which element is most effective?
Decision.
Nehai A – podiya vidmovi.
We introduce a system of hypotheses H1 – the view of the first element, H2 – the view of another element, H3 – the view of the third element.

We know the validity of the hypotheses:
P(H1) = 3/(3+2+5) = 0.3

P(H2) = 2/(3+2+5) = 0.2
P(H3) = 5/(3+2+5) = 0.5

It is necessary to achieve the required level of mental stability and to develop:
P(A|H1) = 0.95, P(A|H2) = 0.9, P(A|H3) = 0.6
a) Find out the reliability of the detection of intelligence in the robot system.
P(A) = P(H1)*P(A|H1) + P(H2)*P(A|H2) + P(H3)*P(A|H3) = 0.3*0.95 + 0.2*0.9 + 0.5 *0.6 = 0.765

b) In the minds of this department, a robotic system was revealed to Vidmov.

The first drawer contains 12 lamps, 1 of which is non-standard;

Which element is most effective? P1 = P (H1) * P (A | H1) / P (A) = 0.3 * 0.95 / 0.765 = 0.373

P2 = P (H2) * P (A | H2) / P (A) = 0.2 * 0.9 / 0.765 = 0.235

P3 = P (H3) * P (A | H3) / P (A) = 0.5 * 0.6 / 0.765 = 0.392

Maximum reliability of the third element. . Bayes' theorem - One of the main theorems of the elementary theory of certainties, which means the certainty of the presence of ideas in the minds, if, on the basis of caution, little information about the ideas is known. With the Bayes formula, one can more accurately overestimate the possibilities, taking into account both previously available information and giving new precautions. “Physical change” and terminology Bayes' formula allows you to “rearrange cause and effect”: given a given fact, you can calculate the credibility of what was caused by this cause. The ideas that represent these “reasons” in this case are called, so it stinks - fortune telling Bayes' formula allows you to “rearrange cause and effect”: given a given fact, you can calculate the credibility of what was caused by this cause. what caused it.).

The insane fairness of the hypothesis is called

a priori (How incredible is the reason burst into flames ), and I’m thinking about the fact that it was revealed -).

An important legacy of the Bayes formula is the formula for the complete reliability of the concept, which lies in

kіlkoh black coolies. b) the other university is faulty;, As it happened, it became difficult for reasons like that.

Behind the Bayes formula

By transfer P(3. At urn A) to the right we remove the shukaniy viraz.

Spam filtering method

The method, based on Bayes' theorem, is more successful in spam filtering.

Description

When you start a filter for the skin of the word in the sheets, this “vaga” is calculated and saved - the probability that the sheet with this word is spam (in the simplest case - behind the classic values: “appearing in spa mі/appearing everything").

When turning over the sheet that again arrives, the probability of the fact that it is spam is calculated, following the established formula for the impersonality of hypotheses. P(3. At urn A | black coolies. b) the other university is faulty;) Once “hypothesis” has a word, and the word “reliability of a hypothesis” has a word, and “the validity of a hypothesis”

- The “vaga” word was counted earlier.

The “vaga” of the sheet in this episode is nothing more than the average “vaga” of all its words.

The updating of the list to “spam” and “non-spam” is carried out by the one who exceeds his “vaga” according to the bar set by the judge (ask to take 60-80%).

After praising a solution from a sheet, “tags” are updated in the database for the words that have been entered. Characteristic This method is simple (elementary algorithms), manual (allows you to do without “black lists” and similar ad hoc techniques), effective (after starting to build up a large selection, it reduces up to 95-97% of spam, even if -What kind of pardons can you give) . Zagalom is all the indications for this ubiquitous search engine, which has its place in practice - on its basis almost all current spam filters are based. However, the method may have some shortcomings in principle:

based on stewed

, what

some words appear more often in spam messages, while others appear more often in regular lists

, and ineffective, because it is not done correctly.

However, as practice shows, such spam is not sent to people “on sight” - only by those who have read the sheet and understood its replacement.

• Another, unimportant, short-term connection with implementation - the method works only with text.
• Paul Graham.

A plan for spam (English).

// Personal website of Paul Graham.

Wikimedia Foundation. 2010.

Marvel at the “Bayes Formula” in other dictionaries:

The formula looks like: de a1, A2,..., Ap nonsense, Zagalna scheme of stastosuvannya F. v.

r.: how can this be done in different ways?

minds that have generated hypotheses A1, A2, ..., An with the well-known facts P(A1),… …

Geological encyclopedia

Allows you to calculate the feasibility of a certain idea through the mental feasibility of this idea of ​​certain hypotheses, and the possibility of these hypotheses.

The formula Nehai is given a unique expanse, and a whole group in pairs... Wikipedia

Allows you to calculate the feasibility of a certain idea through the mental feasibility of this idea of ​​certain hypotheses, and the possibility of these hypotheses.

The formula “Hey” is given an incredible expanse, and another group of ideas, such... ... Wikipedia

- (or the Bayes formula) is one of the main theorems of the theory of certainty, which allows one to determine the certainty of the fact that some idea (hypothesis) has been established for the obviousness of indirect evidence (data), which may be inaccurate... In ikepedia

• The theory of virtuosity and mathematical statistics in science.

On the 360th order, Borzikh D.A. Prote the main emphasis is divided into the prerequisites of average complexity. This has been divided up in order to encourage students to...

Let's create black coolies. again the group b) the other university is faulty;= 1, 2,…, I would like one of them to be sure to be the result of an experiment and a couple of absurd ones.) . Let's say it's okay It is possible that only one in a number of pairs of crazy events will occur at once, creating a new group. :

We will call the pods ( n

a priori certainties Know the certainty of the steps: A – the part will be rejected; black coolies. I’ll finish drinking (a priori).

The reliability of the appearance of the subdivision A is indicated by the formula

full confidence

Butt 16 There are three urns.
The first urn has 5 white and 3 black bags, the other has 4 white and 4 black bags, and the third has 8 white bags.

a priori certainties One of the urns is naturally selected (this could mean, for example, that you are choosing from an additional urn, where there are three boxes numbered 1, 2 and 3). black coolies. There is a sack looming around this urn.
How certain is it that he will show up as black?

- It's pulled out black and blue.

If it were known from which urn the money was drawn, then the credibility could be calculated according to the classical values ​​of credibility. We introduce assumptions (hypotheses) about the fact that the urn is equipped for the purpose of collecting the coolies. Repair and self-development The bag can be pulled either from the first urn (hypothesis), or from another (hypothesis), or from the third (hypothesis).

So, however, there are still chances to choose from the urn, then. happened, then you can change the validity of hypotheses .

The following is what follows: An average of 50% of sick people need to go to a specialized medicine Before, 30% - sick L, 20 % –
from illnesses M. The importance of complete treatment of illness K Lі M more than 0.7 for illness The importance of complete treatment of illness.

a priori certainties The level of reliability is consistent with 0.8 and 0.9. Before L The sick person who, having entered the doctor’s office, will be healthy. M.

Find out the reality of what this ailment is suffering from Repair and self-development Let us introduce the hypotheses: – sick sufferer to sickness

, – ill suffering from illness

Todi is behind the brainwashing department.

Let's introduce it- A sick person who, having entered the doctor’s office, will be healthy.

a priori certainties Behind the washroom

The formula for complete reliability includes: Repair and self-development Behind the Bayes formula.

Butt 19.

Let go of the five cults and all the rumors about the number of white cults of equal strength in the urn. A bag was taken from the urn, and it turned out white.

a priori certainties butt. Repair and self-development What kind of speculation about the cob warehouse of the urn is the most popular?

Let it be - the hypothesis that the urn has white bags, i.e., you can earn six allowances.

Todi is behind the brainwashing department. Repair and self-development Let's introduce it

- a white bag was taken by surprise.

Countable.