Solve an automatic line that produces batteries. Learning to solve problems in probability theory in the Unified State Examination in mathematics

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1. 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 0.95. The probability that the system will mistakenly reject a working battery is 0.01. Find the probability that a battery randomly selected from the package will be rejected.

A battery can be rejected in 2 cases:

1) The battery is faulty. In this case, the probability of its rejection

2) The battery is OK. In this case, the probability of its erroneous rejection

Since the events “battery is good” and “battery is faulty” are incompatible, the probability that a battery randomly selected from the package will be rejected

2. A mechanical watch with a twelve-hour dial broke down at some point and stopped running. Find the probability that the hour hand freezes, reaching the 9 mark, but not reaching the 3 mark.

This sector makes up half of the dial, so the probability is 0.5.

3. In the Magic Land there are two types of weather: good and excellent, and the weather, having established itself in the morning, remains unchanged all day. It is known that with probability 0.9 the weather tomorrow will be the same as today. On June 24, the weather in the Magic Land is good. Find the probability that the weather will be great in Fairyland on June 27th.

Rhor = 0.9, Rotle = 0.1

The probability of excellent weather can be found in a simpler way:

4. A bus runs daily from the district center to the village. The probability that there will be fewer than 23 passengers on the bus on Monday is 0.88. The probability that there will be fewer than 14 passengers is 0.49. Find the probability that the number of passengers will be from 14 to 22.

The probability that the number of passengers will be from 14 to 22 is equal to the product of the probabilities of 2 events:

1) The number of passengers will be greater than or equal to 14, i.e. 1 – 0.49 = 0.51

2) The number of passengers will be less than 23, i.e. 0.88

5. Based on customer reviews, Mikhail Mikhailovich assessed the reliability of two online stores. The probability that the desired product will be delivered from store A is 0.85. The probability that this product will be delivered from store B is 0.87. Mikhail Mikhailovich ordered goods from both stores at once. Assuming that online stores operate independently of each other, find the probability that no store will deliver the product.

6. To enter the institute for the specialty “Translator”, an applicant must score at least 75 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 75 points in each of three subjects – mathematics, Russian language and social studies.

The probability that applicant I. will receive at least 75 points in mathematics is 0.9, in Russian - 0.6, in a foreign language - 0.8 and in social studies - 0.6.

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

To be admitted to one of the specialties, an applicant must pass an exam in mathematics and Russian language and foreign language or social studies.

7. The probability that a student P. will correctly solve more than 7 problems on a history test is 0.58. The probability that P. will correctly solve more than 6 problems is 0.64. Find the probability that P. will solve exactly 7 problems correctly.

8. When manufacturing bearings with a diameter of 74 mm, the probability that the diameter will differ from the specified one by less than 0.01 mm is 0.986. Find the probability that a random bearing will have a diameter less than 73.99 mm or greater than 74.01 mm.

9. The probability that a new vacuum cleaner will be repaired under warranty within a year is 0.09. In a certain city, out of 1,000 vacuum cleaners sold during the year, 97 units were received by the warranty workshop. How different is the frequency of the “warranty repair” event from its probability in this city?

Event frequency “warranty repair” = 97/1000 = 0.097

0,097 - 0,09 = 0,007

10. There are 21 students in the class, among them two friends - Oleg and Sergey. The class is randomly divided into three equal groups. Find the probability that Oleg and Sergey will be in the same group.

11. In a certain city, out of 2000 babies born, 1070 are boys. Find the frequency of births of girls in this city. Round the result to the nearest thousand.

12. To advance to the next round of competition, a football team needs to score at least 9 points in two games. If a team wins, it receives 6 points, if there is a draw, 3 points, and if it loses, 0 points. Find the probability that the team will advance to the next round of the competition. Consider that in each game the probabilities of winning and losing are the same and equal to 0.3.

Advancement to the next round is possible with two possible outcomes of two games:

1) Two victories.

2) Win and draw

Probability of a draw 1 - 0.3 - 0.3 = 0.4

Since both options are incompatible, then

13. 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 Russia will perform after a group from Germany and after a group from China? Round the result to the nearest hundredth.

There are 3 possible options:

1) Russia before China and Germany (China and Germany in all variants - in any order).

2) Russia between China and Germany.

3) Russia after China and Germany.

14. Cowboy John hits a fly on the wall with a probability of 0.9 if he shoots with a zeroed revolver. If John shoots from an unsighted revolver, he hits a fly with a probability of 0.1. There are 10 revolvers on the table, only two of which have been shot. Cowboy John sees a fly on the wall, randomly grabs the first revolver he comes across and shoots the fly. Find the probability that John misses.

Probability of a miss from a targeted weapon 1 - 0.9 = 0.1

Probability of a miss from an unshooted weapon 1 – 0.1 = 0.9

The probability of choosing a sighted weapon is 0.2, an unsighted one is 0.8

15. An agricultural company purchases chicken eggs from two households. 55% of eggs from the first farm are eggs of the highest category, and from the second farm - 45% of eggs of the highest category. In total, 50% of eggs receive the highest category. Find the probability that an egg purchased from this agricultural company will come from the first farm.

Let's denote:

x1 – number of eggs from 1 farm.

x2 – number of eggs from 2 farms.

Total number of eggs y = x1 + x2

Then:

0.55x1 + 0.45x2 = 0.5y

0.45x1 + 0.55x2 = 0.5y

Subtract the second from the first equation:

0.1x1 – 0.1x2 = 0

Therefore x1 = x2, i.e. Both farms produce the same number of eggs, so the required probability is 0.5.

16. The probability that a new personal computer will last more than a year is 0.9. The probability that it will last more than two years is 0.83. Find the probability that it will last less than two years but more than a year.

17. The room is illuminated by a lantern with three lamps. The probability of one lamp burning out within a year is 0.23. Find the probability that at least one lamp will not burn out during the year.

Let's find the probability of the opposite event - all three lamps will burn out within a year.

Then the probability of the opposite event (at least one lamp does not burn out)

18. A biathlete shoots at targets 8 times. The probability of hitting the target with one shot is 0.5. Find the probability that the biathlete hit the targets the first 4 times and missed the last 4 times. Round the result to the nearest hundredth.

There are problems with rounding to the nearest hundredth...

19. 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 0.3. The probability that both machines will run out of coffee is 0.16. Find the probability that at the end of the day there will be coffee left in both machines.

Probability that the second machine has run out of coffee

The likelihood is that by the end of the day there will be coffee left in both machines.

0.327

20. At a geometry exam, a student gets one question from the list of exam questions. The probability that this is a Trigonometry question is 0.3. The probability that this is an inscribed circle question is 0.25. 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.

It follows from the condition that the presence of a question on one of the named topics is an incompatible event with the presence of a question on the second topic, therefore

21. Two factories produce the same glass for car headlights. The first factory produces 35% of these glasses, the second - 65%. The first factory produces 4% of defective glass, and the second – 2%. Find the probability that glass accidentally purchased in a store will be defective.

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“The automatic line produces batteries. The probability that a finished battery is faulty is 0.02. Before packaging, each battery goes through a control system. » — 22 tasks found

Task B6 ()

(views: 199 , answers: 3 )

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 0.96. The probability that the system will mistakenly reject a working battery is 0.05. Find the probability that a randomly selected manufactured battery will be rejected by the inspection system.

Task B6 ()

(views: 207 , answers: 3 )

An automatic line produces batteries. The probability that a finished battery is faulty is 0.03. Before packaging, each battery goes through a control system. The probability that the system will reject a faulty battery is 0.99. The probability that the system will mistakenly reject a working battery is 0.02. Find the probability that a randomly selected manufactured battery will be rejected by the inspection system.