Option 1.

A random event associated with some experience is understood as any event that, during the implementation of this experience

a) cannot happen;

b) either it happens or it doesn’t;

c) will definitely happen.

If the event A occurs if and only if an event occurs IN, then they are called

a) equivalent;

b) joint;

c) simultaneous;

d) identical.

If a complete system consists of 2 incompatible events, then such events are called

a) opposite;

b) incompatible;

c) impossible;

d) equivalent.

A 1 – appearance of an even number of points. Event A 2 - appearance of 2 points. Event A 1  A 2 is what fell

a) 2; b) 4; at 6; d) 5.

The probability of a reliable event is equal to

a) 0; b) 1; at 2; d) 3.

Probability of the product of two dependent events A And IN calculated by the formula

a) P(AB) = P(A)P(B); b) P(AB) = P(A)+P(B) – P(A) P(B);

c) P(A B) = P(A)+P(B) + P(A) P(B); d) P(A B) = P(A) P(A | B).

From 25 exam tickets, numbered from 1 to 25, a student draws 1 at random. What is the probability that the student will pass the exam if he knows the answers to 23 tickets?

A) ; b) ; V) ; G) .

There are 10 balls in a box: 3 white, 4 black, 3 blue. 1 ball was pulled out at random. What is the probability that it will be either white or black?

A) ; b) ; V) ; G) .

There are 2 drawers. The first contains 5 standard and 1 non-standard parts. The second contains 8 standard and 2 non-standard parts. One part is taken out at random from each box. What is the probability that the removed parts will be standard?

A) ; b) ; V) ; G) .

From the word " mathematics"One letter is selected at random. What is the probability that this letter " A»?

A) b) ; V) ; G) .

Option 4.

If an event cannot occur in a given experience, then it is called

a) impossible;

b) incompatible;

c) optional;

d) unreliable.

Experiment with tossing dice. Event A the number of points not exceeding 3 is rolled. Event IN an even number of points is rolled. Event A IN is that the side with the number fell out

a) 1; b) 2; at 3; d) 4.

Events that form a complete system of pairwise incompatible and equally probable events are called

a) elementary;

b) incompatible;

c) impossible;

d) reliable.

a) 0; b) 1; at 2; d) 3.

The store received 30 refrigerators. 5 of them have a manufacturing defect. One refrigerator is selected at random. What is the probability that it will be without a defect?

A) ; b); V) ; G) .

Probability of the product of two independent events A And IN calculated by the formula

a) P(A B) = P(A) P(B | A); b) P(AB) = P(A) + P(B) – P(A) P(B);

c) P(AB) = P(A) + P(B) + P(A) P(B); d) P(AB) = P(A)P(B).

There are 20 people in the class. Of these, 5 are excellent students, 9 are good students, 3 have C grades and 3 have B grades. What is the probability that a randomly selected student is either an excellent student or an excellent student?

A) ; b) ; V) ; G) .

9. The first box contains 2 white and 3 black balls. The second box contains 4 white and 5 black balls. One ball is drawn at random from each box. What is the probability that both balls are white?

A) ; b) ; V) ; G) .

10. The probability of a certain event is equal to

a) 0; b) 1; at 2; d) 3.

Option 3.

If in a given experiment no two of the events can occur simultaneously, then such events are called

a) incompatible;

b) impossible;

c) equivalent;

d) joint.

A set of incompatible events such that at least one of them must occur as a result of the experiment is called

a) an incomplete system of events; b) a complete system of events;

c) a holistic system of events; d) not a holistic system of events.

By producing events A 1 And A 2

a) an event occurs A 1 , event A 2 not happening;

b) an event occurs A 2 , event A 1 not happening;

c) events A 1 And A 2 occur simultaneously.

In a batch of 100 parts, 3 are defective. What is the probability that a part chosen at random will be defective?

A)
; b) ; V)
;
.

The sum of the probabilities of events forming a complete system is equal to

a) 0; b) 1; at 2; d) 3.

The probability of an impossible event is

a) 0; b) 1; at 2; d) 3.

A And IN calculated by the formula

a) P(A+B) = P(A) + P(B); b) P(A+B) = P(A) + P(B) – P(AB);

c) P(A+B) = P(A) + P(B) + P(AB); d) P(A+B) = P(AB) – P(A) + P(B).

There are 10 textbooks arranged in random order on a shelf. Of these, 1 is in mathematics, 2 in chemistry, 3 in biology and 4 in geography. The student randomly took 1 textbook. What is the probability that it will be in either mathematics or chemistry?

A) ; b) ; V) ; G) .

a) incompatible;

b) independent;

c) impossible;

d) dependent.

Two boxes contain pencils of the same size and shape. In the first box: 5 red, 2 blue and 1 black pencils. In the second box: 3 red, 1 blue and 2 yellow. One pencil is drawn at random from each box. What is the probability that both pencils will be blue?

A) ; b) ; V) ; G) .

Option 2.

If an event necessarily occurs in a given experience, then it is called

a) joint;

b) real;

c) reliable;

d) impossible.

If the occurrence of one of the events does not exclude the occurrence of another in the same trial, then such events are called

a) joint;

b) incompatible;

c) dependent;

d) independent.

If the occurrence of event B does not have any effect on the probability of the occurrence of event A, and vice versa, the occurrence of event A does not have any effect on the probability of the occurrence of event B, then events A and B are called

a) incompatible;

b) independent;

c) impossible;

d) dependent.

The sum of events A 1 And A 2 is an event that occurs when

a) at least one of the events occurs A 1 or A 2 ;

b) events A 1 And A 2 do not occur;

c) events A 1 And A 2 occur simultaneously.

The probability of any event is a non-negative number not exceeding

a) 1; b) 2; at 3; d) 4.

From the word " automation"One letter is selected at random. What is the probability that it will be the letter " A»?

A) ; b) ; V) ; G) .

Probability of the sum of two incompatible events A And IN calculated by the formula

a) P(A+B) = P(A) + P(B); b) P(A+B) = P(AB) – P(A) + P(B);

c) P(A+B) = P(A) + P(B) + P(AB); d) P(A+B) = P(A) + P(B) – P(AB).

The first box contains 2 white and 5 black balls. The second box contains 2 white and 3 black balls. One ball was drawn at random from each box. What is the probability that both balls are black?

A) ; b) ; V) ; G) .

Exercise

Demo option

1. and - independent events. Then the following statement is true: a) they are mutually exclusive events

b)

G)

d)

2. , , - probability of events , , 0 " style="margin-left:55.05pt;border-collapse:collapse;border:none">

3. Probabilities of events and https://pandia.ru/text/78/195/images/image012_30.gif" width="105" height="28 src=">.gif" width="55" height="24"> There is:

a) 1.25 b) 0.3886 c) 0.25 d) 0.8614

d) there is no correct answer

4. Prove the equality using truth tables or show that it is false.

Section 2. Probabilities of combining and intersecting events, conditional probability, formulas of total probability and Bayes.

Exercise: Choose the correct answer and mark the corresponding letter in the table.

Demo option

1. We throw two dice at the same time. What is the probability that the sum of the points drawn is not more than 6?

A) ; b) ; V) ; G) ;

d) there is no correct answer

2. Each letter of the word CRAFT is written on a separate card, then the cards are shuffled. We take out three cards at random. What is the probability of receiving the word "FOREST"?

A) ; b) ; V) ; G) ;

d) there is no correct answer

3. Among second-year students, 50% never missed classes, 40% missed classes no more than 5 days per semester, and 10% missed classes for 6 or more days. Among the students who did not miss classes, 40% received the highest score, among those who missed no more than 5 days - 30%, and among the remaining - 10% received the highest score. The student received the highest score on the exam. Find the probability that he missed classes for more than 6 days.

a) https://pandia.ru/text/78/195/images/image024_14.gif" width="17 height=53" height="53">; c) ; d) ; e) there is no correct answer

Test on the course of probability theory and mathematical statistics.

Section 3. Discrete random variables and their numerical characteristics.

Exercise: Choose the correct answer and mark the corresponding letter in the table.

Demo option

1 . Discrete random variables X and Y are specified by their own laws

distribution

Random variable Z = X+Y. Find probability

a) 0.7; b) 0.84; c) 0.65; d) 0.78; d) there is no correct answer

2. X, Y, Z are independent discrete random variables. The value X is distributed according to the binomial law with parameters n=20 and p=0.1. The Y value is distributed according to a geometric law with the parameter p=0.4. The value of Z is distributed according to Poisson's law with parameter =2. Find the variance of the random variable U= 3X+4Y-2Z

a) 16.4 b) 68.2; c) 97.3; d) 84.2; d) there is no correct answer

3. Two-dimensional random vector (X, Y) defined by the distribution law

Event, event . What is the probability of event A+B?

a) 0.62; b) 0.44; c) 0.72; d) 0.58; d) there is no correct answer

Test on the course of probability theory and mathematical statistics.

Section 4. Continuous random variables and their numerical characteristics.

Exercise: Choose the correct answer and mark the corresponding letter in the table.

Option demo

1. Independent continuous random variables X and Y are uniformly distributed on the segments: X on https://pandia.ru/text/78/195/images/image032_6.gif" width="32" height="23">.

Random variable Z = 3X +3Y +2. Find D(Z)

a) 47.75; b) 45.75; c) 15.25; d) 17.25; d) there is no correct answer

2 ..gif" width="97" height="23">

a) 0.5; b) 1; c) 0; d) 0.75; d) there is no correct answer

3. A continuous random variable X is specified by its probability density https://pandia.ru/text/78/195/images/image036_7.gif" width="99" height="23 src=">.

a) 0.125; b) 0.875; c)0.625; d) 0.5; d) there is no correct answer

4. The random variable X is normally distributed with parameters 8 and 3. Find

a) 0.212; b) 0.1295; c)0.3413; d) 0.625; d) there is no correct answer

Test on the course of probability theory and mathematical statistics.

Section 5. Introduction to mathematical statistics.

Exercise: Choose the correct answer and mark the corresponding letter in the table.

Demo option

1. The following estimates of the mathematical expectation are proposed https://pandia.ru/text/78/195/images/image041_6.gif" width="98" height="22">:

A) https://pandia.ru/text/78/195/images/image043_5.gif" width="205" height="40">

B) https://pandia.ru/text/78/195/images/image045_4.gif" width="205" height="40">

D) 0 " style="margin-left:69.2pt;border-collapse:collapse;border:none">

2. The variance of each measurement in the previous problem is . Then the most efficient of the unbiased estimates obtained in the first problem will be the estimate

3. Based on the results of independent observations of a random variable X that obeys Poisson's law, construct an estimate of the unknown parameter using the method of moments 425 " style="width:318.65pt;margin-left:154.25pt;border-collapse:collapse; border:none">

a) 2.77; b) 2.90; c) 0.34; d) 0.682; d) there is no correct answer

4. Half-width of the 90% confidence interval constructed to estimate the unknown mathematical expectation of a normally distributed random variable X for a sample size n=120, sample mean https://pandia.ru/text/78/195/images/image052_3.gif" width="19 " height="16">=5, yes

a) 0.89; b) 0.49; c) 0.75; d) 0.98; d) there is no correct answer

Validation matrix – test demo

Section 1
		A-	B+	IN-	G-	D+
Section 2
Section 3.
Section 4
Section 5

OPTION 1

1.In a random experiment, two dice are thrown. Find the probability that the total will be 5 points. Round the result to hundredths.

2. In a random experiment, a symmetrical coin is tossed three times. Find the probability of getting heads exactly twice.

3. On average, out of 1,400 garden pumps on sale, 7 leak. Find the probability that one pump randomly selected for control does not leak.

4. The competition of performers is held over 3 days. A total of 50 performances have been announced - one from each country. There are 34 performances on the first day, the rest are distributed equally between the remaining days. The order of performances is determined by drawing lots. What is the probability that a Russian representative will perform on the third day of the competition?

5. The taxi company has 50 cars; 27 of them are black with yellow inscriptions on the sides, the rest are yellow with black inscriptions. Find the probability that a yellow car with black lettering will respond to a random call.

6. Bands perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a group from Germany will perform after a group from France and after a group from Russia? Round the result to hundredths.

7. What is the probability that a randomly selected natural number from 41 to 56 is divisible by 2?

8. In the collection of mathematics tickets there are only 20 tickets, 11 of them contain a question on logarithms. Find the probability that a student will get a question on logarithms on a randomly selected exam ticket.

9. The picture shows a labyrinth. The spider crawls into the maze at the Entrance point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path along which it has not yet crawled. Considering the choice of the further path to be random, determine with what probability the spider will come to the exit.

10. To enter the institute for the specialty "Translator", an applicant must score at least 79 points on the Unified State Examination in each of three subjects - mathematics, Russian language and a foreign language. To enroll in the specialty “Customs Affairs”, you need to score at least 79 points in each of three subjects - mathematics, Russian language and social studies.

The probability that applicant B. will receive at least 79 points in mathematics is 0.9, in Russian - 0.7, in a foreign language - 0.8 and in social studies - 0.9.

OPTION 2

1. There are three sellers in the store. Each of them is busy with a client with probability 0.3. Find the probability that at a random moment in time all three sellers are busy at the same time (assume that customers come in independently of each other).

2. In a random experiment, a symmetrical coin is tossed three times. Find the probability that the RRR outcome will occur (heads all three times).

3. The factory produces bags. On average, for every 200 quality bags, there are four bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to hundredths.

4. The competition of performers is held over 3 days. A total of 55 performances have been announced - one from each country. There are 33 performances on the first day, the rest are distributed equally between the remaining days. The order of performances is determined by drawing lots. What is the probability that a Russian representative will perform on the third day of the competition?

5. There are 10 digits on the telephone keypad, from 0 to 9. What is the probability that a randomly pressed digit will be less than 4?

6. A biathlete shoots at targets 9 times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hits the targets the first 3 times and misses the last six times. Round the result to hundredths.

7. Two factories produce identical glasses for car headlights. The first factory produces 30 of these glasses, the second - 70. The first factory produces 4 defective glasses, and the second - 1. Find the probability that a glass accidentally bought in a store will be defective.

8. In the collection of tickets for chemistry there are only 25 tickets, 6 of them contain a question on hydrocarbons. Find the probability that a student will get a question on hydrocarbons on a randomly selected exam ticket.

9. To enter the institute for the specialty "Translator", an applicant must score at least 69 points on the Unified State Exam in each of three subjects - mathematics, Russian language and a foreign language. To enroll in the “Management” specialty, you need to score at least 69 points in each of three subjects - mathematics, Russian language and social studies.

The probability that applicant T. will receive at least 69 points in mathematics is 0.6, in Russian - 0.6, in a foreign language - 0.5 and in social studies - 0.6.

Find the probability that T. will be able to enroll in one of the two mentioned specialties.

10. The picture shows a labyrinth. The spider crawls into the maze at the Entrance point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path along which it has not yet crawled. Considering the choice of the further path to be random, determine with what probability the spider will come to the exit.

OPTION 3

1. 60 athletes are participating in the gymnastics championship: 14 from Hungary, 25 from Romania, the rest from Bulgaria. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete competing first is from Bulgaria.

2. An automatic line produces batteries. The probability that a finished battery is faulty is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a faulty battery is 0.97. The probability that the system will mistakenly reject a working battery is 0.02. Find the probability that a battery randomly selected from the package will be rejected.

3. To enter the institute for the specialty “International Relations”, the applicant must score at least 68 points on the Unified State Examination in each of three subjects - mathematics, Russian language and a foreign language. To enroll in the Sociology specialty, you need to score at least 68 points in each of three subjects - mathematics, Russian language and social studies.

The probability that applicant V. will receive at least 68 points in mathematics is 0.7, in Russian - 0.6, in a foreign language - 0.6 and in social studies - 0.7.

Find the probability that V. will be able to enroll in one of the two mentioned specialties.

4. The picture shows a labyrinth. The spider crawls into the maze at the Entrance point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path along which it has not yet crawled. Considering the choice of the further path to be random, determine with what probability the spider will come to the exit.

5. What is the probability that a randomly selected natural number from 52 to 67 is divisible by 4?

6. At the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.1. The probability that this is a Trigonometry question is 0.35. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics in the exam.

7. Seva, Slava, Anya, Andrey, Misha, Igor, Nadya and Karina cast lots as to who should start the game. Find the probability that the boy will start the game.

8. 5 scientists from Spain, 4 from Denmark and 7 from Holland came to the seminar. The order of reports is determined by drawing lots. Find the probability that the twelfth report will be a report by a scientist from Denmark.

9. In the collection of tickets on philosophy there are only 25 tickets, 8 of them contain a question on Pythagoras. Find the probability that a student will not get a question on Pythagoras on a randomly selected exam ticket.

10. There are two payment machines in the store. Each of them can be faulty with probability 0.09, regardless of the other machine. Find the probability that at least one machine is working.

OPTION 4

1. Bands perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a group from the USA will perform after a group from Vietnam and after a group from Sweden? Round the result to hundredths.

2. The probability that student T will solve more than 8 problems correctly on a history test is 0.58. The probability that T. will correctly solve more than 7 problems is 0.64. Find the probability that T. will solve exactly 8 problems correctly.

3. The factory produces bags. On average, for every 60 quality bags, there are six bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to hundredths.

4. Sasha had four candies in his pocket - “Mishka”, “Vzlyotnaya”, “Belochka” and “Grilyazh”, as well as the keys to the apartment. While taking out the keys, Sasha accidentally dropped one piece of candy from his pocket. Find the probability that the “Vzlyotnaya” candy is lost.

5. The picture shows a labyrinth. The spider crawls into the maze at the Entrance point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path along which it has not yet crawled. Considering the choice of the further path to be random, determine with what probability the spider will come to the exit.

6. In a random experiment, three dice are rolled. Find the probability that the total will be 15 points. Round the result to hundredths.

7. A biathlete shoots at targets 10 times. The probability of hitting the target with one shot is 0.7. Find the probability that the biathlete hit the targets the first 7 times and missed the last three. Round the result to hundredths.

8. 5 scientists from Switzerland, 7 from Poland and 2 from Great Britain came to the seminar. The order of reports is determined by drawing lots. Find the probability that the thirteenth report will be a report by a scientist from Poland.

9. To enter the institute for the specialty “International Law”, an applicant must score at least 68 points on the Unified State Examination in each of three subjects - mathematics, Russian language and a foreign language. To enroll in the Sociology specialty, you need to score at least 68 points in each of three subjects - mathematics, Russian language and social studies.

The probability that applicant B. will receive at least 68 points in mathematics is 0.6, in Russian - 0.8, in a foreign language - 0.5 and in social studies - 0.7.

Find the probability that B. will be able to enroll in one of the two mentioned specialties.

10. In a shopping center, two identical machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.25. The probability that both machines will run out of coffee is 0.14. Find the probability that at the end of the day there will be coffee left in both machines.

1. MATHEMATICAL SCIENCE ESTABLISHING THE REGULARITIES OF RANDOM PHENOMENA IS:

a) medical statistics

b) probability theory

c) medical demography

d) higher mathematics

Correct answer: b

2. THE POSSIBILITY OF REALIZING ANY EVENT IS:

a) experiment

b) case diagram

c) regularity

d) probability

The correct answer is d

3. EXPERIMENT IS:

a) the process of accumulation of empirical knowledge

b) the process of measuring or observing an action for the purpose of collecting data

c) study covering the entire population of observation units

d) mathematical modeling of reality processes

The correct answer is b

4. OUTCOME IN THE THEORY OF PROBABILITY IS UNDERSTANDED:

a) uncertain result of the experiment

b) a certain result of the experiment

c) dynamics of the probabilistic process

d) the ratio of the number of observation units to the general population

The correct answer is b

5. SAMPLING SPACE IN PROBABILITY THEORY IS:

a) structure of the phenomenon

b) all possible outcomes of the experiment

c) the relationship between two independent populations

d) the relationship between two dependent populations

The correct answer is b

6. A FACT THAT MAY OR NOT HAPPEN IF A CERTAIN SET OF CONDITIONS IS IMPLEMENTED:

a) frequency of occurrence

b) probability

c) phenomenon

d) event

The correct answer is d

7. EVENTS THAT HAPPEN WITH THE SAME FREQUENCY AND NONE OF THEM IS OBJECTIVELY MORE POSSIBLE THAN THE OTHERS:

a) random

b) equally probable

c) equivalent

d) selective

The correct answer is b

8. AN EVENT WHICH WILL DEFINITELY HAPPEN IF CERTAIN CONDITIONS ARE REALIZED IS CONSIDERED:

a) necessary

b) expected

c) reliable

d) priority

The correct answer is in

8. THE OPPOSITE OF A RELIABLE EVENT IS THE EVENT:

a) unnecessary

b) unexpected

c) impossible

d) non-priority

The correct answer is in

10. PROBABILITY OF A RANDOM EVENT APPEARING:

a) greater than zero and less than one

b) more than one

c) less than zero

d) represented by integers

The correct answer is a

11. EVENTS FORM A COMPLETE GROUP OF EVENTS IF CERTAIN CONDITIONS ARE REALIZED, AT LEAST ONE OF THEM:

a) will certainly appear

b) appears in 90% of experiments

c) appears in 95% of experiments

d) appears in 99% of experiments

The correct answer is a

12. THE PROBABILITY OF THE APPEARANCE OF ANY EVENT FROM THE COMPLETE GROUP OF EVENTS WHEN CERTAIN CONDITIONS ARE IMPLEMENTED IS EQUAL:

The correct answer is d

13. IF NO TWO EVENTS WHEN CERTAIN CONDITIONS ARE REALIZED CAN APPEAR AT THE SAME TIME, THEN THEY ARE CALLED:

a) reliable

b) incompatible

c) random

d) probable

The correct answer is b

14. IF, UNDER CERTAIN CONDITIONS, NONE OF THE EVENTS ASSESSED IS OBJECTIVELY MORE POSSIBLE THAN THE OTHERS, THEN THEY ARE:

a) equal

b) joint

c) equally possible

d) incompatible

The correct answer is in

15. A QUANTITY WHICH CAN TAKE DIFFERENT VALUES DUE TO CERTAIN CONDITIONS IS CALLED:

a) random

b) equally possible

c) selective

d) total

The correct answer is a

16. IF WE KNOW THE NUMBER OF POSSIBLE OUTCOMES OF SOME EVENT AND THE TOTAL NUMBER OF OUTCOMES IN THE SAMPLE SPACE, THEN WE CAN CALCULATE:

a) conditional probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is b

17. WHEN WE DO NOT HAVE SUFFICIENT INFORMATION ABOUT WHAT IS HAPPENING AND CANNOT DETERMINE THE NUMBER OF POSSIBLE OUTCOMES OF AN EVENT INTERESTING US, WE CAN CALCULATE:

a) conditional probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is in

18. BASED ON YOUR PERSONAL OBSERVATIONS, YOU OPERATE:

a) objective probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is d

19. THE SUM OF TWO EVENTS A AND IN EVENT CALLED:

a) consisting of the sequential occurrence of either event A or event B, excluding their joint occurrence

b) consisting in the occurrence of either event A or event B

c) consisting in the occurrence of either event A, or event B, or events A and B together

d) consisting in the occurrence of event A and event B together

The correct answer is in

20. BY THE PRODUCT OF TWO EVENTS A AND IN IS AN EVENT CONSISTED OF:

a) the joint occurrence of events A and B

b) sequential occurrence of events A and B

c) the occurrence of either event A, or event B, or events A and B together

d) the occurrence of either event A or event B

The correct answer is a

21. IF EVENT A DOES NOT AFFECT THE PROBABILITY OF AN EVENT OCCURING IN, AND ON THE CONVERSE, THEY CAN BE CONSIDERED:

a) independent

b) ungrouped

c) remote

d) heterogeneous

The correct answer is a

22. IF EVENT A INFLUENCES THE PROBABILITY OF AN EVENT OCCURING IN, AND ON THE CONVERSE, THEY CAN BE CONSIDERED:

a) homogeneous

b) grouped

c) instantaneous

d) dependent

The correct answer is d

23. THEOREM OF ADDING PROBABILITIES:

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

b) the probability of the sequential occurrence of two joint events is equal to the sum of the probabilities of these events

c) the probability of the sum of two incompatible events is equal to the sum of the probabilities of these events

d) the probability of the non-occurrence of two incompatible events is equal to the sum of the probabilities of these events

The correct answer is in

24. ACCORDING TO THE LAW OF LARGE NUMBERS, WHEN AN EXPERIMENT IS CARRIED OUT A LARGE NUMBER OF TIMES:

a) empirical probability tends to classical

b) empirical probability moves away from the classical one

c) subjective probability exceeds classical

d) empirical probability does not change in relation to the classical one

The correct answer is a

25. PROBABILITY OF TWO EVENTS OCCURING A AND IN EQUAL TO THE PRODUCT OF THE PROBABILITY OF ONE OF THEM ( A) ON THE CONDITIONAL PROBABILITY OF OTHER ( IN), CALCULATED UNDER THE CONDITION THAT THE FIRST TOOK PLACE:

a) probability multiplication theorem

b) the theorem of addition of probabilities

c) Bayes' theorem

d) Bernoulli's theorem

The correct answer is a

26. ONE OF THE CONSEQUENCES OF THE PROBABILITY MULTIPLICATION THEOREM:

b) if event A affects event B, then event B also affects event A

d) if event Ane affects event B, then event B does not affect event A

The correct answer is in

27. ONE OF THE CONSEQUENCES OF THE PROBABILITY MULTIPLICATION THEOREM:

a) if event A depends on event B, then event B also depends on event A

b) the probability of producing independent events is equal to the product of the probabilities of these events

c) if event A does not depend on event B, then event B does not depend on event A

d) the probability of producing dependent events is equal to the product of the probabilities of these events

The correct answer is b

28. THE INITIAL PROBABILITIES OF HYPOTHESES BEFORE RECEIVING ADDITIONAL INFORMATION ARE CALLED

a) a priori

b) a posteriori

c) preliminary

d) initial

The correct answer is a

29. PROBABILITIES REVISED AFTER RECEIVING ADDITIONAL INFORMATION ARE CALLED

a) a priori

b) a posteriori

c) preliminary

d) final

The correct answer is b

30. WHAT THEOREM OF PROBABILITY THEORY CAN BE APPLIED WHEN MAKING A DIAGNOSIS

a) Bernoulli

b) Bayesian

c) Chebyshev

d) Poisson

The correct answer is b

1.Specify true definition. The sum of two events is called:

a) A new event, consisting in the fact that both events occur simultaneously;

b) A new event, consisting in the fact that either the first or the second occurs, or both together; +

1. Specify true definition. The product of two events is called:

a) A new event, consisting in the fact that both events occur simultaneously;+

b) A new event, consisting in the fact that either the first or the second occurs, or both together;

c) A new event, consisting in the fact that one thing happens but another does not happen.

1. Specify true definition. The probability of an event is called:

a) The product of the number of outcomes favorable to the occurrence of the event by the total number of outcomes;

b) The sum of the number of outcomes favorable to the occurrence of the event and the total number of outcomes;

c) The ratio of the number of outcomes favorable to the occurrence of an event to the total number of outcomes;+

1. Specify true statement. Probability of an impossible event:

b) equal to zero;+

c) equal to one;

1. Specify true statement. Probability of a certain event:

a) greater than zero and less than one;

b) equal to zero;

c) equal to one;+

1. Specify true property. Probability of a random event:

a) greater than zero and less than one;+

b) equal to zero;

c) equal to one;

1. Specify correct statement:

a) The probability of the sum of events is equal to the sum of the probabilities of these events;

b) The probability of the sum of independent events is equal to the sum of the probabilities of these events;

c) The probability of the sum of incompatible events is equal to the sum of the probabilities of these events;+

1. Specify correct statement:

a) The probability of events occurring is equal to the product of the probabilities of these events;

b) The probability of producing independent events is equal to the product of the probabilities of these events;+

c) The probability of the occurrence of incompatible events is equal to the product of the probabilities of these events;

1. Specify true definition.An event is:

a) Elementary outcome;

b) Space of elementary outcomes;

c) A subset of the set of elementary outcomes.+

1. Specify correct answer. What events are called hypotheses?

a) any pairwise incompatible events;

b) pairwise incompatible events, the combination of which forms a reliable event;+

c) space of elementary events.

1. Specify correct answer Bayes formulas define:

a) a priori probability of the hypothesis,

b) posterior probability of the hypothesis,

c) the probability of the hypothesis.+

1. Specify true property. The distribution function of the random variable X is:

a) non-increasing; b) non-decreasing; +c) of any type.

1. Specify true

a) independent+; b) dependent; c) everyone.

1. Specify true property. The equality is valid for random variables:

a) independent; + b) dependent; c) everyone.

1. Specify correct conclusion. From the fact that the correlation moment for two random variables X and Y is equal to zero:

a) there is no functional relationship between X and Y;

b) the values ​​of X and Y are independent;+

c) there is no linear correlation between X and Y;

1. Specify correct answer. The discrete random variable is specified:

a) indicating its probabilities;

b) indicating its distribution law;+

c) assigning each elementary outcome to correspondence

real number.

1. Specify true definition. The mathematical expectation of a random variable is:

a) initial moment of the first order;+

b) first order central moment;

c) an arbitrary moment of the first order.

1. Specify true definition. The variance of a random variable is:

a) second-order initial moment;

b) central moment of the second order;+

c) an arbitrary second-order moment.

1. Specify faithful formula. Formula for calculating the standard deviation of a random variable:

a) +; b) ; V) .

1. Specify true definition. The distribution mode is:

a) the value of a random variable at which the probability is 0.5;

b) the value of a random variable at which either the probability or the density function reaches its maximum value;+

c) the value of a random variable at which the probability equals 0.

1. Specify faithful formula. The variance of a random variable is calculated using the formula:

1. Specify faithful formula. The normal distribution density of a random variable is determined by the formula:

1. Specify correct answer The mathematical expectation of a random variable distributed according to the normal distribution law is equal to:

1. Specify correct answer. The mathematical expectation of a random variable distributed according to the exponential distribution law is equal to:

1. Specify correct answer. The variance of a random variable distributed according to the exponential distribution law is equal to:

1. Specify faithful formula. For a uniform distribution, the mathematical expectation is determined by the formula:

1. Specify faithful formula. For a uniform distribution, the dispersion is determined by the formula:

1. Specify incorrect statement. Properties of sample variance:

a) if all options are increased by the same number of times, then the variance will increase by the same number of times.

b) the variance of the constant is zero.

c) if all options are increased by the same number, then the sample variance will not change.+

1. Specify true statement. Parameter estimation is called:

a) Presentation of observations as independent random variables having the same distribution law.

b) a set of observation results;

c) any function of observation results.+

1. Specify true statement. Estimates of distribution parameters have the following property:

a) undisplaced;+

b) significance;

c) importance.

1. Specify not true statement.

a) The maximum likelihood method is used to obtain estimates;

b) The sample variance is a biased estimator for the variance;

c) Unbiased, inconsistent, effective estimates are used as statistical estimates of parameters.+

1. Specify incorrect statement. The distribution function of a two-dimensional random variable has the following properties:

A) ; b) ; c) +.

1. Specify incorrect statement:

a) Using a multidimensional distribution function, one can always find one-dimensional (marginal) distributions of individual components.

b) From one-dimensional (marginal) distributions of individual components one can always find a multidimensional distribution function.

c) Using a multidimensional density function, one can always find one-dimensional (marginal) distribution densities of individual components.

1. Specify correct statement. The variance of the difference between two random variables is determined by the formula:

A); b)+; V) .

1. Specify incorrect statement. Formula for calculating joint density:

1. Specify incorrect statement. Random variables X and Y are called independent if:

a) The distribution law of the random variable X does not depend on the value of the random variable Y.

c) the correlation coefficient between random variables X and Y is zero.

1. Specify correct answer. The formula is:

a) an analogue of Bayes’ formula for continuous random variables;

b) an analogue of the total probability formula for continuous random variables;+

c) an analogue of the formula for the product of probabilities of independent events for continuous random variables.

1. Specify incorrect definition:

a) The initial moment of the order of a two-dimensional random variable (X,Y) is the mathematical expectation of the product by, i.e.

b) The central moment of the order of a two-dimensional random variable (X,Y) is the mathematical expectation of the product of centers on, i.e.)

c) The correlation moment of a two-dimensional random variable (X,Y) is the mathematical expectation of the product by, i.e. +

1. Specify correct answer. The variance of a random variable distributed according to the normal distribution law is equal to:

1. Specify incorrect statement. The simplest problems of mathematical statistics are:

a) sampling and grouping of statistical data obtained as a result of the experiment;

b) determination of distribution parameters, the type of which is known in advance;+

c) obtaining an estimate of the probability of the event being studied.