Probability theory and mathematical statistics test doc. Tests in the discipline “Probability Theory and Mathematical Statistics. Topic: Theorems of addition and multiplication of probabilities

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

a) medical statistics

b) probability theory

c) medical demographics

d) higher mathematics

Correct answer: b

2. THE POSSIBILITY OF IMPLEMENTING ANY EVENT IS:

a) experiment

b) scheme of cases

c) regularity

d) probability

The correct answer is g

3. EXPERIMENT IS:

a) the process of accumulation of empirical knowledge

b) the process of measuring or observing an action in order to collect data

c) study covering the entire population of observation units

d) mathematical modeling of reality processes

Correct answer b

4. OUTCOME IN PROBABILITY THEORY IS UNDERSTANDING:

a) an uncertain result of the experiment

b) a certain result of the experiment

c) the dynamics of the probabilistic process

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

Correct answer b

5. SAMPLE SPACE IN PROBABILITY THEORY IS:

a) the structure of the phenomenon

b) all possible outcomes of the experiment

c) the ratio between two independent sets

d) the ratio between two dependent populations

Correct answer b

6. A FACT WHICH MAY OCCUR OR NOT OCCUR IN THE IMPLEMENTATION OF A CERTAIN COMPLEX OF CONDITIONS:

a) frequency of occurrence

b) probability

c) a phenomenon

d) an event

The correct answer is g

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

a) random

b) equiprobable

c) equivalent

d) selective

Correct answer b

8. AN EVENT WHICH WILL NEED TO OCCUR IN THE IMPLEMENTATION OF CERTAIN CONDITIONS IS CONSIDERED:

a) necessary

b) expected

c) reliable

d) priority

Correct answer in

8. THE OPPOSITE OF A CREDIBLE EVENT IS AN EVENT:

a) unnecessary

b) unexpected

c) impossible

d) non-priority

Correct answer in

10. PROBABILITY OF A RANDOM EVENT:

a) greater than zero and less than one

b) more than one

c) less than zero

d) represented by whole numbers

Correct answer a

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

a) will always appear

b) will appear in 90% of experiments

c) will appear in 95% of experiments

d) will appear in 99% of experiments

Correct answer a

12. THE PROBABILITY OF THE APPEARANCE OF ANY EVENT FROM THE FULL GROUP OF EVENTS IN THE IMPLEMENTATION OF CERTAIN CONDITIONS IS EQUAL TO:

The correct answer is g

13. IF NO TWO EVENTS CAN APPEAR SIMULTANEOUSLY DURING THE IMPLEMENTATION OF CERTAIN CONDITIONS, THEY ARE CALLED:

a) credible

b) incompatible

c) random

d) probable

Correct answer b

14. IF NONE OF THE EVALUATED EVENTS IS OBJECTIVELY MORE POSSIBLE THAN THE OTHERS IN THE IMPLEMENTATION OF CERTAIN CONDITIONS, THEN THEY:

a) equal

b) joint

c) equally likely

d) incompatible

Correct answer in

15. A VALUE WHICH CAN TAKE DIFFERENT VALUES UNDER THE IMPLEMENTATION OF CERTAIN CONDITIONS IS CALLED:

a) random

b) equally possible

c) selective

d) total

Correct answer a

16. IF WE KNOW THE NUMBER OF POSSIBLE OUTCOMES OF A 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

Correct answer b

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

a) conditional probability

b) classical probability

c) empirical probability

d) subjective probability

Correct answer in

18. BASED ON YOUR PERSONAL OBSERVATIONS, YOU DO:

a) objective probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is g

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

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

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

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

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

Correct answer in

20. PRODUCTION OF TWO EVENTS BUT AND IN IS AN EVENT CONSISTING IN:

a) the joint occurrence of events A and B

b) consecutive appearance of events A and B

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

d) the occurrence of either event A or event B

Correct answer a

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

a) independent

b) ungrouped

c) remote

d) heterogeneous

Correct answer a

22. IF EVENT BUT AFFECTS THE PROBABILITY OF AN EVENT IN, AND CONVERSUS, THEY CAN BE COUNTERED:

a) homogeneous

b) grouped

c) one-time

d) dependent

The correct answer is g

23. PROBABILITY ADDITION THEOREM:

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 successive 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 non-occurrence of two incompatible events is equal to the sum of the probabilities of these events

Correct answer in

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

a) empirical probability tends to classical

b) the empirical probability moves away from the classical

c) subjective probability exceeds the classical one

d) the empirical probability does not change with respect to the classical

Correct answer a

25. PROBABILITY OF THE PRODUCT OF TWO EVENTS BUT AND IN IS EQUAL TO THE PRODUCT OF THE PROBABILITY OF ONE OF THEM ( BUT) ON THE CONDITIONAL PROBABILITY OF THE OTHER ( IN), CALCULATED UNDER THE CONDITION THAT THE FIRST OCCURRED:

a) probability multiplication theorem

b) probability addition theorem

c) Bayes' theorem

d) Bernoulli's theorem

Correct answer a

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

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

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

Correct answer in

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

a) if event A depends on event B, then event B 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 the product of dependent events is equal to the product of the probabilities of these events

Correct answer b

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

a) a priori

b) a posteriori

c) preliminary

d) initial

Correct answer a

29. PROBABILITIES REVISED AFTER ADDITIONAL INFORMATION IS REVIEWED ARE CALLED

a) a priori

b) a posteriori

c) preliminary

d) final

Correct answer b

30. WHAT THEOREM OF PROBABILITY THEORY CAN BE APPLIED IN THE DIAGNOSIS

a) Bernoulli

b) Bayesian

c) Chebyshev

d) Poisson

Correct answer b

1 option

1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=m=100

2. They threw a dice. What is the probability of getting an even number of points?

Answer:

1 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - all parts are defective.

Answer:

- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and at least one boiler are running.

Answer:

5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 5.

Answer:

6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: all the boys will be in the same subgroup?

7. A coin was flipped 3 times. What is the probability that heads will come up 3 times.

Answers:

8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is white.

Answers:

9. Choose the correct answer:

Answers:

10. Choose the correct answer: Total probability formula

11. Find P (AB) if

Answers:

12. Find if P(A) = 0.2

13. Events A and B are incompatible. Find P(A + B) if P(A) = P(B) = 0.3

14. Find P (A + B) if P (A) \u003d P (B) \u003d 0.3 P (AB) \u003d 0.1

15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 10, m = 2

16. The most probable number of occurrences of an event when repeating tests is found by the formula:

17. The sum of the products of each DSV value and the corresponding probability is called.

p = 0.9; n = 10

22. . The binomial law of distribution of DSV is given. Find P(x

23. Find the appropriate formula: M(x) =?

Answers:

To find .

Answers:

27. A random variable has a uniform distribution if

Answers:

Answer: a) b)

c) d)

30. In formula

Answers:

Test on the subject "Probability Theory and Mathematical Statistics"

Option 2

1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=1000; m=100

Answer: a) 0.75 b) 1 c) 0.5 d) 0.1

2. They threw a dice. What is the probability of getting more than four

Answer:

3. There are 20 standard parts and 7 defective parts in a box. Pulled out three parts. Event A 1 – 1st part is defective, A 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - all parts are standard.

Answer:

4. Let A - the machine works, B- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and at least two boilers are running.

Answer:

5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 8.

Answer:

6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 2 young men will be in one subgroup, and 4 in another?

Answers a) 8 b) 168 c) 840 d) 56

7. A coin was flipped 3 times. What is the probability that heads will come up once.

Answers:

8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is blue.

Answers:

9. Choose the correct answer:

Answers:

10. Choose the correct answer: Bernoulli formula

11. Find P (AB) if

Answers:

12. Find if P(A) = 0.8

Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6

13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.25 P(B) = 0.45

Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6

14. Find P (A + B) if P (A) \u003d 0.2 P (B) \u003d 0.8 P (AB) \u003d 0.1

Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7

15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 20, m = 3

Answers: a) b) 0.2 c) 0.25 d) 0.15

16. Local Moivre-Laplace theorem

17. The mathematical expectation of the square of the difference between the random variable X and its mathematical expectation is called:

Answers: a) the variance of a random variable b) the mathematical expectation of the DSV

C) standard deviation d) DSV distribution law

18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).

p = 0.8; n = 9

Answers: a) 8.4 b) 6 c) 7.2 d) 9

19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).

p = 0.8; n = 9

Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9

20. The binomial law of distribution of DSV is given. Find M(x).

Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8

21. The binomial law of distribution of DSV is given. Find D(x).

Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84

22. The binomial law of distribution of DSV is given. Find P (x > 2).

Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792

23. Find the appropriate formula: D (x) \u003d?

Answers:

24. The law of distribution of DSV is given. Find M(x).

Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9

25. The law of distribution of DSV is set. To find.

Answers:

27. A random variable has a normal distribution if

Answers:

28. Find the differential distribution function f(x)if

Answers:

29. Find the integral distribution function F(x) if

Answer: a) b)

c) d)

30. In formula

Answers:

Test on the subject "Probability Theory and Mathematical Statistics"

3 option

1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=500 m=255

Answer: a) 0.75 b) 1 c) 0.5 d) 0.1

2. They threw a dice. What is the probability of getting less than five

Answer:

4. Let A be a machine, B- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and all boilers are running.

Answer:

5. An n-volume collected works were placed on the shelf in random order. What is the probability that there are 100 booksyat in ascending order of volume numbers if n = 10.

Answer:

6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 3 young men will be in one subgroup, and 3 in another?

Answers a) 8 b) 168 c) 840 d) 56

7. A coin was flipped 3 times. What is the probability that heads will come up at least once.

Answers:

8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is yellow.

Answers:

9. Choose the correct answer:

Answers:

10. Choose the correct answer: Bayss formula

11. Find P (AB) if

Answers:

12. Find if P(A) = 0.5

Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6

13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.7 P(B) = 0.1

Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6

14. Find P (A + B) if P (A) \u003d 0.5 P (B) \u003d 0.2 P (AB) \u003d 0.1

Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7

15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 40, m = 10

Answers: a) b) 0.2 c) 0.25 d) 0.15

16. Laplace integral theorem

17. The square root of the dispersion of a random variable is called:

Answers: a) the variance of a random variable b) the mathematical expectation of the DSV

C) standard deviation d) DSV distribution law

18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).

p = 0.7; n = 12

Answers: a) 8.4 b) 6 c) 7.2 d) 9

19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).

p = 0.7; n = 12

Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9

20. The binomial law of distribution of DSV is given. Find M(x).

Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8

21. The binomial law of distribution of DSV is given. Find D(x).

Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84

22. The binomial law of distribution of DSV is given. Find P(0

Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792

(x) = ?

Answers:

24. The law of distribution of DSV is given. Find M(x).

Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9

25. The law of distribution of DSV is set. To find

Answers:

27. A random variable has an exponential distribution if

Answers:

28. Find the differential distribution function f(x)if

Answers:

29. Find the integral distribution function F(x) if

Answer: a) b)

c) d)

30. In formula

Answers:

Test on the subject "Probability Theory and Mathematical Statistics"

4 option

1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=400 m=300

Answer: a) 0.75 b) 1 c) 0.5 d) 0.1

2. They threw a dice. What is the probability of getting less than six

Answer:

4. Let A be a machine, B- the -th boiler is working ( =1,2,3). Log an event: the unit is running the machine-boiler unit is running if the machine is running; 1st boiler and at least one of the other two boilers.

Answer:

5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 7.

Answer:

6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 5 young men will be in one subgroup, and 1 in another?

Answers a) 8 b) 168 c) 840 d) 56

7. A coin was flipped 3 times. What is the probability that heads will come up more than once.

Answers:

8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is blue.

Answers:

9. Choose the correct answer:

Answers:

10. Choose the correct answer: The formula for the product of the probabilities of dependent events

11. Find P (AB) if

Answers:

12. Find if P(A) = 0.4

Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6

13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.6 P(B) = 0.3

Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6

14. Find P (A + B) if P (A) \u003d 0.6 P (B) \u003d 0.4 P (AB) \u003d 0.4

Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7

15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 60, m = 10

Answers: a) b) 0.2 c) 0.25 d) 0.15

16. Bernoulli's theorem

17. A correspondence that establishes a connection between the possible values of a random variable and their probabilities is called:

Answers: a) the variance of a random variable b) the mathematical expectation of the DSV

C) standard deviation d) DSV distribution law

18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).

p = 0.6; n = 10

Answers: a) 8.4 b) 6 c) 7.2 d) 9

19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).

p = 0.6; n = 10

Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9

20. The binomial law of distribution of DSV is given. Find M(x).

Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8

21. The binomial law of distribution of DSV is given. Find D(x).

Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84

22. . The binomial law of distribution of DSV is given. Find P(1

Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792

23. Find the corresponding formula:

Answers:

24. The law of distribution of DSV is given. Find M(x).

Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9

25. The law of distribution of DSV is set. To find

Answers:

27. A random variable has a binomial distribution if

Answers:

28. Find the differential distribution function f(x)if

Answers:

29. Find the integral distribution function F(x) if

Answer: a) b)

c) d)

30. In formula

Answers:

OPTION 1

1. In a random experiment, two dice are thrown. Find the probability of getting 5 points in total. Round the result to the nearest hundredth.

2. In a random experiment, a symmetrical coin is thrown three times. Find the probability that heads come up exactly twice.

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

4. The competition of performers is held in 3 days. There are 50 entries in total, one from each country. There are 34 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place 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 inscriptions will arrive at a random call.

6. Groups 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 the nearest hundredth.

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

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

9. The figure 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 that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.

10. To enter the institute for the specialty "Translator", the applicant must score at least 79 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Customs", you need to score at least 79 points in each of the 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 a probability of 0.3. Find the probability that at a random moment of time all three sellers are busy at the same time (assume that customers enter independently of each other).

2. In a random experiment, a symmetrical coin is tossed three times. Find the probability that the outcome of the RPP will come (all three times it comes up tails).

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 the nearest hundredth.

4. The competition of performers is held in 3 days. There are 55 entries in total, one from each country. There are 33 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place 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 number will be less than 4?

6. Biathlete shoots at targets 9 times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hit the targets the first 3 times and missed the last 6. Round the result to the nearest hundredth.

7. Two factories produce the same glass 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 randomly bought in a store will be defective.

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

9. To enter the institute for the specialty "Translator", the applicant must score at least 69 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Management", you need to score at least 69 points in each of the 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 enter one of the two specialties mentioned.

10. The figure 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 that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.

OPTION 3

1. 60 athletes participate 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 who competes first is from Bulgaria.

2. Automatic production line for batteries. The probability that a finished battery is defective is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a bad battery is 0.97. The probability that the system will mistakenly reject a good battery is 0.02. Find the probability that a randomly selected battery 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 the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Sociology", you need to score at least 68 points in each of the 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 B. will be able to enter one of the two specialties mentioned.

4. The figure 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 that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.

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

6. On 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 related to these two topics at the same time. Find the probability that the student will get a question on one of these two topics on the exam.

7. Seva, Slava, Anya, Andrey, Misha, Igor, Nadya and Karina cast lots for who to start the game. Find the probability that a 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 a draw. Find the probability that the report of a scientist from Denmark will be the twelfth.

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

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

OPTION 4

1. Groups perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the likelihood that a band from the USA will perform after a band from Vietnam and after a band from Sweden? Round the result to the nearest hundredth.

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

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 the nearest hundredth.

4. Sasha had four sweets in his pocket - “Mishka”, “Vzlyotnaya”, “Squirrel” and “Roasting”, as well as the keys to the apartment. Taking out the keys, Sasha accidentally dropped one candy from his pocket. Find the probability that the take-off candy is lost.

5. The figure 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 that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.

6. In a random experiment, three dice are thrown. Find the probability of getting 15 points in total. Round the result to the nearest hundredth.

7. 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 3. Round the result to the nearest hundredth.

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

9. To enter the institute for the specialty "International Law", the applicant must score at least 68 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Sociology", you need to score at least 68 points in each of the 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 enter one of the two specialties mentioned.

10. There are two identical coffee machines in the mall. 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 by the end of the day there will be coffee left in both vending machines.

Option 1.

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

a) cannot happen

b) either happens or it doesn't;

c) will definitely happen.

If the event BUT occurs when and only when an event occurs IN, then they are called

a) equivalent;

b) joint;

c) simultaneous;

d) identical.

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

a) opposite;

b) incompatible;

c) impossible;

d) equivalent.

BUT 1 - the appearance of an even number of points. Event BUT 2 - the appearance of 2 points. Event BUT 1  BUT 2 is that it fell

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

The probability of a certain event is equal to

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

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

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

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

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

but) ; b) ; in) ; G) .

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

but) ; b) ; in) ; G) .

There are 2 boxes. The first one contains 5 standard and 1 non-standard parts. The second has 8 standard and 2 non-standard parts. One item is drawn at random from each box. What is the probability that the removed parts will be standard?

but) ; b) ; in) ; G) .

From the word " maths One letter is chosen at random. What is the probability that this letter but»?

but) b) ; in) ; G) .

Option 4.

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

a) impossible;

b) incompatible;

c) optional;

d) unreliable.

Experience with throwing a dice. Event BUT no more than 3 points are dropped. Event IN get an even number of points. Event BUT IN consists in the fact that the edge with the number

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

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

a) elementary;

b) incompatible;

c) impossible;

d) reliable.

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

The store received 30 refrigerators. 5 of them have a factory defect. One refrigerator is randomly selected. What is the probability that it will be defect free?

but) ; b); in) ; G) .

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

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

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 triples and 3 have deuces. What is the probability that a randomly selected student is either a good student or an excellent student?

but) ; b) ; in) ; 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?

but) ; b) ; in) ; G) .

10. The probability of a certain event is equal to

a) 0; b) 1; in 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) an integral system of events; d) not an integral system of events.

The product of events BUT 1 And BUT 2

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

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

c) events BUT 1 And BUT 2 are happening at the same time.

In a batch of 100 parts, 3 are defective. What is the probability that a randomly selected item will be defective?

but)
; b) ; in)
;
.

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

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

The probability of an impossible event is

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

BUT And IN calculated by the formula

a) P (A + B) \u003d P (A) + P (B); b) P (A + B) \u003d P (A) + P (B) - P (A B);

c) P(A+B) = P(A) + P(B) + P(A B); d) P (A + B) \u003d P (A B) - P (A) + P (B).

10 textbooks are randomly placed on the shelf. Of these, 1 in mathematics, 2 in chemistry, 3 in biology and 4 in geography. The student randomly took 1 textbook. What is the probability that he will be in either math or chemistry?

but) ; b) ; in) ; 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 pencil. 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 are blue?

but) ; b) ; in) ; 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 occurrence of event A, and vice versa, the occurrence of event A does not have any effect on the probability of occurrence of event B, then events A and B are called

a) incompatible;

b) independent;

c) impossible;

d) dependent.

The sum of events BUT 1 And BUT 2 is an event that takes place when

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

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

c) events BUT 1 And BUT 2 are happening at the same time.

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

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

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

but) ; b) ; in) ; G) .

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

a) P (A + B) \u003d P (A) + P (B); b) P (A + B) \u003d P (A B) - P (A) + P (B);

c) P(A+B) = P(A) + P(B) + P(A B); d) P (A + B) \u003d 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 is drawn at random from each box. What is the probability that both balls are black?

but) ; b) ; in) ; G) .