Probability test for students. Test on probability theory and mathematical statistics on the topics "elements of combinatorics", "foundations of probability theory", "discrete random variables". Topic: Theorems of addition and multiplication of probabilities

Exercise

Demo option

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

b)

G)

e)

2. ,, - event probabilities,, 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

e) there is no correct answer

4. Prove the equality using truth tables or show that it is not true.

Section 2. Probabilities of combination and intersection of events, conditional probability, total probability and Bayesian formulas.

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

Demo option

1. Throw two dice at the same time. What is the probability that the total of the dropped points is not more than 6?

a) ; b); v) ; G) ;

e) 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 getting the word "FOREST"?

a) ; b); v) ; G) ;

e) there is no correct answer

3. Among second-year students, 50% never missed classes, 40% missed classes for 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 mark on the exam. Find the probability that he has been missing 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 given by their own laws

distribution

Random variable Z = X + Y. Find the probability

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

2. X, Y, Z - independent discrete random variables. The quantity X is distributed according to the binomial law with the parameters n = 20 and p = 0.1. The quantity Y is distributed according to the geometric law with the parameter p = 0.4. The Z value 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; e) there is no correct answer

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

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

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

Test for 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 evenly distributed on the segments: X at 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; e) there is no correct answer

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

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

3. A continuous random variable X is given 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; e) there is no correct answer

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

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

Test for 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 ">

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

2. There is a variance of each dimension in the previous problem. Then the most effective 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 obeying Poisson's law, construct by the method of moments an estimate of the unknown parameter 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; e) there is no correct answer

4. Half-width of 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; e) there is no correct answer

Validation Matrix - Test Demo

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

Given to the present moment in the open bank of USE problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is the classical definition of probability.

The easiest way to understand the formula is with examples.
Example 1. The basket contains 9 red balls and 3 blue ones. The balls differ only in color. At random (without looking) we get one of them. What is the probability that the ball chosen in this way will turn out to be blue?

A comment. In problems on the theory of probability, something happens (in this case, our action to pull out the ball), which can have a different result - the outcome. It should be noted that the result can be viewed in different ways. "We pulled out some kind of ball" - also the result. "We pulled out the blue ball" is the result. “We pulled this particular ball out of all possible balls” - this least generalized view of the result is called an elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.

Solution. Now let's calculate the probability of choosing a blue ball.
Event A: "the selected ball turned out to be blue"
The total number of all possible outcomes: 9 + 3 = 12 (the number of all balls that we could pull out)
The number of favorable outcomes for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P (A) = 3/12 = 1/4 = 0.25
Answer: 0.25

Let us calculate the probability of choosing a red ball for the same problem.
The total number of possible outcomes will remain the same, 12. Number of favorable outcomes: 9. Probability sought: 9/12 = 3/4 = 0.75

The probability of any event always lies in the range from 0 to 1.
Sometimes in everyday speech (but not in the theory of probability!) The probability of events is estimated as a percentage. The transition between mathematical and conversational assessment is done by multiplying (or dividing) by 100%.
So,
Moreover, the probability is equal to zero for events that cannot happen - they are incredible. For example, in our example, this would be the probability of pulling a green ball out of the basket. (The number of favorable outcomes is 0, P (A) = 0/12 = 0, if calculated by the formula)
Probability 1 have events that will definitely happen, with no options. For example, the probability that “the selected ball will be either red or blue” is for our problem. (Number of favorable outcomes: 12, P (A) = 12/12 = 1)

We've looked at a classic example to illustrate the definition of probability. All such problems of the exam in probability theory are solved by applying this formula.
In place of red and blue balls, there may be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on a topic (prototypes,), defective and high-quality bags or garden pumps (prototypes,) - the principle remains the same.

They differ slightly in the formulation of the problem of the probability theory of the exam, where you need to calculate the probability of an event occurring on a certain day. (,) As in the previous tasks, you need to determine what is the elementary outcome, and then apply the same formula.

Example 2. The conference lasts three days. On the first and second day 15 speakers will speak, on the third day - 20. What is the probability that Professor M.'s report will fall on the third day, if the order of the reports is determined by drawing lots?

What is the elementary outcome here? - Assignment of a professor's report to one of all possible serial numbers for a speech. The draw is attended by 15 + 15 + 20 = 50 people. Thus, the report of Professor M. can receive one of 50 numbers. This means that there are only 50 elementary outcomes.
What are the favorable outcomes? - Those in which it turns out that the professor will speak on the third day. That is, the last 20 numbers.
According to the formula, the probability P (A) = 20/50 = 2/5 = 4/10 = 0.4
Answer: 0.4

The drawing of lots here is the establishment of a random correspondence between people and ordered places. In example 2, the establishment of correspondence was considered from the point of view of which of the places a particular person could occupy. You can approach the same situation from the other side: which of the people with what probability could get to a specific place (prototypes,,,):

Example 3. The draw involves 5 Germans, 8 French and 3 Estonians. What is the likelihood that the first (/ second / seventh / last - it does not matter) will be a Frenchman.

The number of elementary outcomes is the number of all possible people who could get to a given place by lot. 5 + 8 + 3 = 16 people.
Favorable outcomes - French. 8 people.
Seeking probability: 8/16 = 1/2 = 0.5
Answer: 0.5

The prototype is slightly different. There are some more creative problems about coins () and dice (). The solution to these problems can be seen on the prototype pages.

Here are some examples of throwing a coin or dice.

Example 4. When we flip a coin, what is the probability of getting heads?
Outcomes 2 - heads or tails. (it is believed that the coin never falls on the edge) Favorable outcome - tails, 1.
Probability 1/2 = 0.5
Answer: 0.5.

Example 5. What if we flip a coin twice? What is the probability of hitting heads both times?
The main thing is to determine which elementary outcomes we will consider when flipping two coins. After flipping two coins, one of the following results may be obtained:
1) PP - both times came tails
2) PO - first time tails, second time heads
3) OP - heads first time, tails second time
4) OO - heads both times
There are no other options. Hence, there are 4 elementary outcomes. Favorable of them is only the first, 1.
Probability: 1/4 = 0.25
Answer: 0.25

What is the probability that two coin tosses will come up tails once?
The number of elementary outcomes is the same, 4. Favorable outcomes - the second and third, 2.
Probability of hitting one tails: 2/4 = 0.5

In such tasks, one more formula may come in handy.
If for one toss of a coin we have 2 possible outcomes, then for two tosses the results will be 2 2 = 2 2 = 4 (as in example 5), for three tosses 2 2 2 = 2 3 = 8, for four: 2 · 2 · 2 · 2 = 2 4 = 16, ... for N throws, the possible results will be 2 · 2 · ... · 2 = 2 N.

So, you can find the probability of getting 5 heads out of 5 coin tosses.
The total number of elementary outcomes: 2 5 = 32.
Favorable outcomes: 1. (RRRRR - all 5 tails)
Probability: 1/32 = 0.03125

The same is true for the dice. With one throw, there are 6 possible results here. So, for two throws: 6 6 = 36, for three 6 6 6 = 216, etc.

Example 6. We throw the dice. What is the probability that an even number will be dropped?

Total outcomes: 6, according to the number of faces.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6 = 0.5

Example 7. Throw in two dice. What is the chance of a total of 10 rolled? (round to hundredths)

There are 6 possible outcomes for one die. Hence, for two, according to the above rule, 6 6 = 36.
What outcomes will be favorable for 10 in total?
10 must be decomposed into the sum of two numbers from 1 to 6. This can be done in two ways: 10 = 6 + 4 and 10 = 5 + 5. This means that the following options are possible for cubes:
(6 on the first and 4 on the second)
(4 on the first and 6 on the second)
(5 on the first and 5 on the second)
Total, 3 options. Seeking probability: 3/36 = 1/12 = 0.08
Answer: 0.08

Other types of B6 problems will be covered in one of the following How to Solve articles.

Option number 1

1. There are 14 defective bricks in a batch of 800 bricks. The boy picks one brick at random from this batch and throws it from the eighth floor of the construction site. What is the likelihood that an abandoned brick will be defective?
2. The 11th grade physics examination book consists of 75 tickets. In 12 of them there is a question about lasers. What is the probability that Styop's student, choosing a ticket at random, stumbles upon a question about lasers?
3. 3 athletes from Italy, 5 athletes from Germany and 4 from Russia are taking part in the 100m running championship. The lane number for each athlete is drawn by lot. What is the likelihood that an athlete from Italy will be on the second lane?
4. 1,500 bottles of vodka were brought to the store. It is known that 9 of them are overdue. Find the probability that an alcoholic who chooses one bottle at random ends up buying an expired one.
5. There are 120 offices of various banks in the city. Granny chooses one of these banks at random and opens a deposit in it for 100,000 rubles. It is known that during the crisis 36 banks went bankrupt, and the depositors of these banks lost all their money. What is the likelihood that Granny won't lose her contribution?
6. In one 12-hour shift, a worker produces 600 parts on a numerically controlled 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 piece at random and checks it. What is the probability that he will come across a defective part?

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

Option number 1

1. At the Kievsky railway station in Moscow, there are 28 ticket windows, crowded with 4,000 passengers who want to buy train tickets. Statistically, 1,680 of these passengers are inadequate. Find the probability that the cashier sitting outside the 17th window will find an inadequate passenger (considering that passengers choose the ticket office at random).
2. Russian Standard Bank runs a lottery for its clients - holders of Visa Classic and Visa Gold cards. There will be drawn 6 Opel Astra cars, 1 Porsche Cayenne car and 473 iPhone 4 phones. It is known that the manager Vasya issued a Visa Classic card and became the winner of the lottery. What is the likelihood that he will win an Opel Astra if the prize is chosen at random?
3. A school was repaired in Vladivostok and 1200 new plastic windows were installed. An 11th grade student who did not want to take the USE in mathematics found 45 boulders on the lawn and started throwing them at the windows at random. As a result, he broke 45 windows. Find the probability that the window in the director's office will not be broken.
4. A US military plant has received a batch of 9,000 fake Chinese-made microcircuits. These microcircuits are installed in electronic sights for the M-16 rifle. It is known that 8766 microcircuits in the specified batch are faulty, and sights with such microcircuits will not work correctly. Find the probability that the randomly selected electronic sight works correctly.
5. Granny keeps 2,400 cucumber jars in the attic of her country house. It is known that 870 of them have long gone rotten. When her granddaughters came to see grandma, she presented him with one jar from her collection, choosing it at random. What is the likelihood that the granddaughter received a jar of rotten cucumbers?
6. A team of 7 migrant builders offers apartment renovation services. During the summer season, they completed 360 orders, and in 234 cases they did not remove construction waste from the entrance. Utilities pick one apartment at random and check the quality of the renovation work. Find the likelihood that utility workers will not stumble upon building debris while checking.

Answers:

Var # 1
answer	0,0175	0,16	0,25	0,006		0,015
Option number 2
answer	0,42	0,0125	0,9625	0,026	0,3625	0,35

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

a) medical statistics

b) probability theory

c) medical demography

d) higher mathematics

Correct answer: b

2. THE POSSIBILITY OF IMPLEMENTING ANY EVENT IS:

a) experiment

b) case diagram

c) regularity

d) probability

The correct answer is d

3. EXPERIMENT IS:

a) the process of accumulating empirical knowledge

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

c) study covering the entire general population of observation units

d) mathematical modeling of reality processes

The correct answer is b

4. BY THE OUTCOME IN THE THEORY OF PROBABILITY UNDERSTAND:

a) indeterminate experimental result

b) a certain result of the experiment

c) the dynamics of the probabilistic process

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

The correct answer is b

5. SELECTED SPACE IN THE THEORY OF PROBABILITY IS:

a) the structure of the phenomenon

b) all possible outcomes of the experiment

c) the relationship between two independent aggregates

d) the relationship between two dependent populations

The correct answer is b

6. A FACT WHICH COULD HAPPEN OR NOT HAPPEN IN THE IMPLEMENTATION OF A CERTAIN SET OF CONDITIONS:

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 ARE OBJECTIVELY MORE POSSIBLE THAN OTHERS:

a) random

b) equiprobable

c) equivalent

d) selective

The correct answer is b

8. AN EVENT WHICH WILL HAPPEN IN THE IMPLEMENTATION OF CERTAIN CONDITIONS IS CONSIDERED:

a) necessary

b) expected

c) reliable

d) priority

Correct answer in

8. THE OPPOSITE TO A TRUE EVENT IS THE EVENT:

a) unnecessary

b) unexpected

c) impossible

d) non-priority

Correct answer in

10. PROBABILITY OF THE APPEARANCE OF A RANDOM EVENT:

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 FULL GROUP OF EVENTS IF DURING THE IMPLEMENTATION OF CERTAIN CONDITIONS, AT LEAST ONE OF THEM:

a) will certainly appear

b) will appear in 90% of experiments

c) will appear 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 FULL GROUP OF EVENTS DURING THE IMPLEMENTATION OF CERTAIN CONDITIONS IS EQUAL TO:

The correct answer is d

13. IF ANY TWO EVENTS DURING THE IMPLEMENTATION OF CERTAIN CONDITIONS CANNOT APPEAR SIMULTANEOUSLY, THEY ARE CALLED:

a) reliable

b) inconsistent

c) random

d) probable

The correct answer is b

14. IF DURING THE IMPLEMENTATION OF CERTAIN CONDITIONS NONE OF THE EVALUATED EVENTS ARE OBJECTIVELY MORE POSSIBLE THAN OTHERS, THEY:

a) equal

b) joint

c) equally possible

d) incompatible

Correct answer in

15. THE VALUE THAT CAN TAKE DIFFERENT VALUES WHEN IMPLEMENTING 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 A SELECTED SPACE, THEN IT IS POSSIBLE TO 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 OCCURRING AND CANNOT DETERMINE THE NUMBER OF POSSIBLE OUTCOMES OF THE EVENT OF INTERESTING TO 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 OPERATE:

a) objective probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is d

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

a) consisting in the sequential appearance 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. BY PRODUCING TWO EVENTS A AND V IS AN EVENT CONSISTING IN:

a) the joint occurrence of events A and B

b) sequential occurrence of events A and B

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

d) the appearance of either event A or event B

The correct answer is a

21. IF EVENT A DOES NOT AFFECT THE PROBABILITY OF THE EVENT V, AND VERSATELY, THIS CAN BE COUNTED:

a) independent

b) ungrouped

c) remote

d) dissimilar

The correct answer is a

22. IF EVENT A AFFECTS THE PROBABILITY OF AN EVENT V, And vice versa, THIS CAN BE COUNTED:

a) homogeneous

b) grouped

c) one-time

d) dependent

The correct answer is d

23. THE THEOREM OF ADDITION OF 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 consecutive 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 PERFORMED A LARGE NUMBER OF TIMES:

a) empirical probability tends to the classical

b) empirical probability moves away from the classical

c) the subjective probability exceeds the classical

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

The correct answer is a

25. PROBABILITY OF TWO EVENTS A AND V IS EQUAL TO THE PRODUCT OF THE PROBABILITY OF ONE OF THEM ( A) FOR THE CONDITIONAL PROBABILITY OF ANOTHER ( V) CALCULATED UNDER THE CONDITION THAT THE FIRST HAS BEEN PLACE:

a) probability multiplication theorem

b) addition theorem for probabilities

c) Bayes' theorem

d) Bernoulli's theorem

The correct answer is a

26. ONE OF THE CONSEQUENCES OF THE MULTIPLICATION OF PROBABILITIES:

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

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

Correct answer in

27. ONE OF THE CONSEQUENCES OF THE MULTIPLICATION OF PROBABILITIES:

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

b) the probability of the product of 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

The correct answer is b

28. INITIAL PROBABILITIES OF HYPOTHESIS BEFORE RECEIVING ADDITIONAL INFORMATION, 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 THE THEORY OF PROBABILITY CAN BE APPLIED IN THE FORMULATION OF DIAGNOSIS

a) Bernoulli

b) Bayesian

c) Chebyshev

d) Poisson

The correct answer is b

OPTION 1

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

2. In a random experiment, a symmetrical coin is thrown three times. Find the probability that it will be heads exactly two times.

3.On average of 1400 garden pumps on sale, 7 are leaking. Find the probability that one pump randomly selected to monitor is not leaking.

4. The competition of performers is held in 3 days. A total of 50 performances are announced - one from each country. On the first day there are 34 performances, the rest are divided equally between the remaining days. The order of performances is determined by drawing lots. What is the probability that the speech of the Russian representative will take place on the third day of the competition?

5. The taxi company has 50 passenger 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 for 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 likelihood 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 is 41 to 56 divisible by 2?

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

9. The figure shows a maze. The spider crawls into the labyrinth 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 further way 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 exam 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 the Russian language - 0.7, 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 in time all three salespeople are busy at the same time (assume that customers come in independently of each other).

2. In a random experiment, a symmetrical coin is thrown three times. Find the probability that the outcome is PPP (it comes up tails all three times).

3. The factory produces bags. On average, there are four bags with hidden defects for every 200 quality bags. Find the likelihood that the bag you buy will be of good quality. Round the result to the nearest hundredth.

4. The competition of performers is held in 3 days. A total of 55 performances have been announced - one from each country. On the first day there are 33 performances, the rest are divided equally between the remaining days. The order of performances is determined by drawing lots. What is the probability that the speech of the Russian representative will take place on the third day of the competition?

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

6. The biathlete shoots targets 9 times. The probability of hitting a target with one shot is 0.8. Find the probability that the biathlete has hit the targets the first 3 times and missed the last six. 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 the glass you accidentally bought in the store turns out to be defective.

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

9. To enter the institute for the specialty "Translator", the applicant must score at least 69 points on the exam 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 an applicant T. will receive at least 69 points in mathematics is 0.6, in the Russian language - 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 figure shows a maze. The spider crawls into the labyrinth 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 to be 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 first athlete comes from Bulgaria.

2. An automatic line makes batteries. The probability that a finished battery is defective is 0.02. Before packing, 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 mistakenly rejects a good battery is 0.02. Find the likelihood 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 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 the Russian language - 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 above-mentioned specialties.

4. The figure shows a maze. The spider crawls into the labyrinth 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 to be 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 exam in geometry, 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 on the exam.

7. Seva, Slava, Anya, Andrey, Misha, Igor, Nadya and Karina threw lots - who should start the game. Find the probability that a boy should start the game.

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

9. In the collection of tickets on philosophy there are only 25 tickets, in 8 of them there is a question about Pythagoras. Find the probability that a student will not get the Pythagoras question on a ticket randomly selected on 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 machine. Find the probability that at least one machine is operational.

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 likelihood that the US group will perform after the Vietnamese group and after the Swedish group? Round the result to the nearest hundredth.

2. The probability that student T. will correctly solve more than 8 problems on the 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, there are six bags with hidden defects for 60 quality bags. Find the likelihood that the bag you buy will be of good quality. Round the result to the nearest hundredth.

4. Sasha had four sweets in his pocket - Mishka, Vzlyotnaya, Squirrel and Grill, 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 Takeoff candy is lost.

5. The figure shows a maze. The spider crawls into the labyrinth 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 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 the nearest hundredth.

7. The biathlete shoots at targets 10 times. The probability of hitting a 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 the nearest hundredth.

8. The seminar was attended by 5 scientists from Switzerland, 7 from Poland and 2 from Great Britain. The order of the reports is determined by drawing lots. Find the probability that the thirteenth will be 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 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 B. will receive at least 68 points in mathematics is 0.6, in the Russian language - 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 vending 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 likelihood that coffee will remain in both machines by the end of the day.