MFP tests on the theory of probability. Tests in the discipline “Probability Theory and Mathematical Statistics. Topic: Theorems of addition and multiplication of probabilities

TEST #1

Topic: Types of random events, classical definition of probability,

elements of combinatorics.

You are offered 5 test items on the topic types of random events, the classical definition of probability, elements of combinatorics. Among the suggested answers only one is true.

TEST #2

Topic: Theorems of addition and multiplication of probabilities.

You are offered 5 test tasks on the topic of the theorem of addition and multiplication of probabilities. Among the suggested answers only one is true.

TEST #3

Topic: Random independent trials according to the Bernoulli scheme.

You are offered 5 test tasks on the topic of random independent tests according to the Bernoulli scheme. Among the suggested answers only one is true.

TEST #4

Topic: One-dimensional random variables.

You are offered 5 test tasks on the topic of one-dimensional random variables, their ways of setting and numerical characteristics. Among the suggested answers only one is true.

Option number 1

1. There are 14 defective bricks in a batch of 800 bricks. The boy chooses at random one brick from this batch and throws it from the eighth floor of the construction site. What is the probability that a thrown brick will be defective?
2. The physics examination book for grade 11 consists of 75 tickets. In 12 of them there is a question about lasers. What is the probability that Step's student, choosing a ticket at random, will stumble upon a question about lasers?
3. 3 athletes from Italy, 5 athletes from Germany and 4 from Russia compete at the 100m championship. The lane number for each athlete is determined by a draw. What is the probability that an athlete from Italy will be on the second lane?
4. 1500 bottles of vodka were delivered to the store. It is known that 9 of them are overdue. Find the probability that an alcoholic who chooses one bottle at random will end up buying the expired one.
5. There are 120 offices of various banks in the city. Grandma chooses one of these banks at random and opens a deposit of 100,000 rubles in it. It is known that during the crisis, 36 banks went bankrupt, and the depositors of these banks lost all their money. What is the probability that Granny will not lose her deposit?
6. In one 12-hour shift, a worker produces 600 parts on a CNC machine. Due to a defect in the cutting tool, 9 defective parts were received on the machine. At the end of the working day, the workshop foreman takes one part at random and checks it. What is the probability that he will get exactly the defective part?

Test on the topic: "Probability theory in the tasks of the exam"

Option number 1

1. At the Kievsky railway station in Moscow, there are 28 ticket windows, next to which 4,000 passengers are crowding, wishing to buy train tickets. According to statistics, 1680 of these passengers are inadequate. Find the probability that the cashier sitting behind the 17th window will encounter an inadequate passenger (taking into account that passengers choose the cashier at random).
2. Russian Standard Bank holds a lottery for its customers - holders of Visa Classic and Visa Gold cards. 6 Opel Astra cars, 1 Porsche Cayenne car and 473 iPhone 4 phones will be raffled off. It is known that the manager Vasya issued a Visa Classic card and became the winner of the lottery. What is the probability that he will win an Opel Astra if the prize is chosen at random?
3. In Vladivostok, a school was renovated and 1,200 new plastic windows were installed. An 11th grade student who did not want to take the USE in mathematics found 45 cobblestones on the lawn and started throwing them at the windows at random. In the end, he broke 45 windows. Find the probability that the window in the director's office is not broken.
4. A batch of 9,000 counterfeit Chinese-made chips has arrived at an American military factory. These microcircuits are installed in electronic sights for the M-16 rifle. 8766 ICs are known to be defective in this batch and scopes with such ICs will not function correctly. Find the probability that a randomly selected electronic sight works correctly.
5. Granny keeps 2,400 jars of cucumbers in the attic of her country house. It is known that 870 of them have long been rotten. When the granddaughter came to the granny, she gave him one jar from her collection, choosing it at random. What is the probability that the granddaughter received a jar of rotten cucumbers?
6. A team of 7 migrant construction workers offers apartment renovation services. During the summer season, they completed 360 orders, and in 234 cases they did not remove construction debris from the entrance. Public utilities choose one apartment at random and check the quality of the repair work. Find the probability that utility workers will not stumble upon building debris when checking.

Answers:

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.

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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.

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9. Choose the correct answer:

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10. Choose the correct answer: Total probability formula

11. Find P (AB) if

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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

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:

Find .

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27. Random value has a uniform distribution if

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Answer: a) b)

c) d)

30. In formula

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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.

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9. Choose the correct answer:

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10. Choose the correct answer: Bernoulli formula

11. Find P (AB) if

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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. Mathematical expectation of the square of the difference between the random variable X and its mathematical expectation 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. Find.

Answers:

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27. A random variable has normal distribution, If

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

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

Answer: a) b)

c) d)

30. In formula

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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:

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 - at least one part is defective.

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:

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10. Choose the correct answer: Bayss formula

11. Find P (AB) if

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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. Find

Answers:

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27. A random variable has an exponential distribution if

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

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

Answer: a) b)

c) d)

30. In formula

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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:

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 - one part is defective and two are standard.

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:

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

11. Find P (AB) if

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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. Find

Answers:

Answers:

27. A random variable has a binomial distribution if

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

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

Answer: a) b)

c) d)

30. In formula

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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 selected natural number 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 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 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. In order 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 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 relationships”, the applicant must score at least 68 points on the exam in each of the three subjects - mathematics, Russian 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.

Basic concepts on the topic:

1. Trial, elementary outcome, trial outcome, event.

2. Certain event, impossible event, random event.

3. Joint events, incompatible events, equivalent events, equally possible events, the only possible events.

4. Complete group of events, opposite events.

5. Elementary event, composite event.

6. The sum of several events, the product of several events. Their geometric interpretation

1. In the problem “Two shots are fired at the target. Find the probability that the target will be hit once" by the test is:

1) * two shots are fired at the target;

2) the target will be hit once;

3) the target will be hit twice.

2. Throw a coin. Event: A - “the coat of arms will fall out”. The event - “a number will come up” is:

1) random;

2) reliable;

3) impossible;

4) * opposite.

3. A dice is rolled. Let's denote the events: A - "loss of 6 points", B - "loss of 4 points", D - "loss of 2 points", C - "loss of an even number of points". Then the event C is

1)
;

2)
;

3)*
;

4)
.

4. The student must pass two exams. Event A - "the student passed the first exam", event B - "the student passed the second exam", event C - "the student passed both exams". Then the event C is

1)*
;

2)
;

3)
;

4)
.

5. From the letters of the word "TASK" one letter is randomly selected. The event - "the letter K is selected" is

1) random;

2) reliable;

3)* impossible;

4) opposite.

6. From the letters of the word "WORLD" one letter is randomly selected. The event - "the letter M is selected" is

1)* random;

2) reliable;

3) impossible.

7. The event - "a white ball is drawn from an urn containing only white balls" is

1) random;

2) * reliable;

3) impossible.

8. Two students take an exam. Events: A - "the first student will pass the exam", B - "the second student will pass the exam" are

1) incompatible;

2) reliable;

3) impossible;

4)*joint.

9. Events are called incompatible if

4) * the onset of one excludes the possibility of the appearance of the other.

10. Events are called the only possible ones if

1) the occurrence of one does not exclude the possibility of the appearance of another;

2) in the implementation of a set of conditions, each of them has an equal opportunity to occur;

3) * during the test, at least one of them will definitely occur;

Topic 2. Classical definition of probability

Basic concepts on the topic:

1. The probability of an event, the classical definition of the probability of a random event.

2. An outcome favorable to the event.

3. Geometric definition of probability.

4. Relative frequency of the event.

5. Statistical definition of probability.

6. Properties of probability.

7. Methods for counting the number of elementary outcomes: permutations, combinations, placements.

Application of all these concepts on practical examples.

Sample test tasks offered in this topic:

1. Events are called equally likely if

1) they are incompatible;

2) * in the implementation of a set of conditions, each of them has an equal opportunity to occur;

3) during the test, at least one of them will definitely occur;

4) the occurrence of one excludes the possibility of the appearance of the other.

2. Test - "throw two coins." Event - "at least one of the coins will have a coat of arms." The number of elementary outcomes that favor this event is equal to:

4) four.

3. Test - "throw two coins." Event - "a coat of arms will fall on one of the coins." The number of all elementary, equally possible, the only possible, incompatible outcomes is equal to:

4)* four.

4. There are 12 balls in the urn, they do not differ in anything except the color. Among these balls, 5 are black and 7 are white. The event is "a white ball is randomly drawn." For this event, the number of favorable outcomes is:

5. There are 12 balls in the urn, they do not differ in anything except the color. Among these balls, 5 are black and 7 are white. The event is "a white ball is randomly drawn." For this event, the number of all outcomes is:

6. The probability of an event takes any value from the interval:

3)
;

4)
;

5)*
.

7. The subscriber forgot the last two digits of the telephone number and, knowing only that they are different, dialed them at random. In how many ways can he do this?

1);

2)*;