Tasks on the theory of probabilities in the form of test. Test at the rate of the theory of probability and mathematical statistics

1.Cow faithful Definition. The system of two events is called:

a) a new event that occurs both events at the same time;

b) a new event consisting in what is happening or first, or second, or both together; +

1. Point faithfuldetermining. The manufacture of two events is called:

a) a new event consistent with both events at the same time; +

b) a new event consisting of what is happening or first, or second, or both together;

c) a new event consisting in what happens one thing but does not happen.

1. Point faithfuldefinition. The event of an event is called:

a) the work of the number of outcomes, conducive to the emergence of an event for the total number of outcomes;

b) the amount of the number of outcomes, conducive to the emergence of an event and the total number of outcomes;

c) the attitude of the number of outcomes that conducive events to the total number of outcomes; +

1. Point faithfulstatement. Probability of the Impossible Event:

b) equal to zero; +

c) is equal to one;

1. Point faithfulstatement. Probability reliable event:

a) more zero and less than one;

b) equal to zero;

c) is equal to unity; +

1. Point faithfulproperty. Probability of a random event:

a) more zero and less than one; +

b) equal to zero;

c) is equal to one;

1. Point rightstatement:

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

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

c) the probability of the amount of incomplete events is equal to the sum of the probabilities of these events; +

1. Point rightstatement:

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

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

c) the likelihood of a work of incompatible events is equal to the product of the probabilities of these events;

1. Point faithfuldefinition This is:

a) elementary outcome;

b) the space of elementary outcomes;

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

1. Point correctanswer. What events are called hypotheses?.

a) any pairs of incomplete events;

b) pairs of incomplete events, whose association forms a reliable event; +

c) space of elementary events.

1. Point correctthe answer of the Bayes formula is determined by:

a) a priori probability of hypothesis,

b) a posteriority probability of hypothesis,

c) probability of hypothesis. +

1. Point faithfulproperty. Distribution function random variable X is:

a) incentive; b) inconsiderable; + c) arbitrary species.

1. Point faithful

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

1. Point faithfulproperty. Equality is true for random variables:

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

1. Point rightconclusion. The fact that the correlation moment for two random variables x and y is zero follows:

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

b) the values \u200b\u200bof x and y are independent; +

b) absent linear correlation between x and y;

1. Point correctanswer. The discrete random variable is asked:

a) indicating its probabilities;

b) indicating its distribution law; +

c) putting each elementary outcome in line with

a valid number.

1. Point faithfuldefinition. The mathematical expectation of a random variable is:

a) the initial moment of the first order; +

b) the central moment of the first order;

c) arbitrary moment of first order.

1. Point faithfuldefinition. The dispersion of a random variable is:

a) the initial moment of the second order;

b) the central moment of the second order; +

c) an arbitrary moment of the second order.

1. Point loyalformula. Formula for calculating the average quadratic deviation of a random variable:

a) +; b); in) .

1. Point faithfuldefinition. Fashion distribution - This:

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

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

c) the value of a random variable in which the probability is 0.

1. Point loyalformula. The dispersion of a random variable is calculated by the formula:

1. Point loyalformula. Density normal distribution Random variance is determined by the formula:

1. Point correctthe answer mathematical expectation of the random variable distributed according to the normal distribution law is:

1. Point correctanswer. The mathematical expectation of the random variable distributed according to the accurate distribution law is:

1. Point correctanswer. The case of a random variable distributed in terms of the indicative distribution law is equal to:

1. Point loyalformula. For a uniform distribution, the mathematical expectation is determined by the formula:

1. Point loyalformula. For uniform distribution, the dispersion is determined by the formula:

1. Point invalidstatement. Selective dispersion properties:

a) If all options increase in the same number of times, then the dispersion will increase at the same number times.

b) Dispersion of constant equals zero.

c) if all options increase on the same number, then the selective dispersion will not change. +

1. Point faithfulstatement. An estimate of the parameters is called:

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

b) a set of observation results;

c) all the function of observation results. +

1. Point faithfulstatement. Estimates of distribution parameters have a property:

a) failure; +

b) significance;

c) importance.

1. Specify nonsense faithfulstatement.

a) the maximum truthful method is used to obtain estimates;

b) the selective dispersion is a dispersion dispersion;

c) Understandable, insolvent, effective assessments are used as statistical estimates of parameters. +

1. Point invalidstatement. For the distribution function of a two-dimensional random variable, properties:

but) ; b); c) +.

1. Point invalidstatement:

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

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

c) on a multidimensional density function can always be found one-dimensional (marginal) distribution density of individual components.

1. Point rightstatement. The dispersion of the difference of two random variables is determined by the formula:

but); b) +; in) .

1. Point invalidstatement. Formula for calculating joint density:

1. Point invalidstatement. Random variables X and Y are called independent if:

a) the law of the distribution of a random value x does not depend on what value the random value y was taken.

c) the correlation coefficient between random values \u200b\u200bX and Y is zero.

1. Point correctanswer. The formula is:

a) analogue of the Bayes formula for continuous random variables;

b) analog formula full probability For continuous random variables; +

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

1. Point invaliddefinition:

a) the initial point of the order of the two-dimensional random variable (X, Y) is called the mathematical expectation of the work on, i.e.

b) the central point of the order of the two-dimensional random variable (X, Y) is called the mathematical expectation of the work of the centered on, i.e.)

c) the correlation torque of the two-dimensional random variable (X, Y) is called the mathematical expectation of the work on, i.e. +.

1. Point correctanswer. The dispersion of the random variable distributed according to the normal distribution law is equal to:

1. Point invalidstatement. The simplest tasks of mathematical statistics are:

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

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

c) obtaining an assessment of the likelihood of the event being studied.

Option 1.

Under random eventassociated with some experience means any event that in the implementation of this experience

a) can not happen;

b) either happening or not;

b) will definitely happen.

If the event BUT happens then and only when the event occurs INthen they are called

a) equivalent;

b) joint;

c) simultaneous;

d) identical.

If the full system consists of 2 inconspicuous events, such events are called

a) opposite;

b) incomplete;

c) impossible;

d) equivalent.

BUT 1 - The emergence of an even number of points. Event BUT 2 - The appearance of 2 points. Event BUT 1  BUT 2 It is that

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

The probability of a reliable event is equal

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

The probability of the work of two dependent events BUTand IN Calculated by formula

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

c) p (a c) \u003d p (a) + p (c) + p (a)  p (c); d) p (a c) \u003d p (a)  P (a | c).

Of the 25 exam tickets, dominated by numbers from 1 to 25, the student of the Namadach extracts 1. What is the likelihood that the student will pass the exam, if he knows the answers to 23 tickets?

but) ; b) ; in) ; d) .

In a box of 10 balls: 3 white, 4 black, 3 blue. Muadach pulled 1 ball. What is the probability that it will be either white or black?

but) ; b) ; in) ; d) .

There are 2 drawers. In the first 5 standard and 1 non-standard detail. In the second 8 standard and 2 non-standard details. From each drawer, the muddy takes out one piece. What is the likelihood that the removal items will be standard?

but) ; b); in) ; d).

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

but) b) ; in) ; d) .

Option 4.

If an event in this experience can not happen, it is called

a) impossible;

b) incomplete;

c) optional;

d) unreliable.

Experience with thrust bone. Event BUT The number of points falls no more 3. Event IN There is an even number of points. Event BUT INlies in the fact that the line fell

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

Events that form a full system in pairs of inconsistencies and equilibious events are called

a) elementary;

b) incomplete;

c) impossible;

d) reliable.

a) 0; b) 1; at 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 without a defect?

but) ; b); in) ; d) .

The probability of the work of two independent events BUT and INcalculated by formula

a) p (a c) \u003d p (a)  P (in | a); b) p (a c) \u003d p (a) + p (c) - p (a)  P (B);

c) p (a c) \u003d p (a) + p (c) + p (a)  p (c); d) p (a c) \u003d p (a)  P (B).

In class 20 people. Of these, 5 excellent students, 9 gooders, 3 have three and 3 have two. What is the likelihood that the chosen chance of a student is either a good one or an excellent student?

but) ; b) ; in) ; d) .

9. In the first box 2 white and 3 black bowls. In the second box 4 white and 5 black balls. Ruadach remove from each box one ball. What is the likelihood that both balls will be white?

but) ; b) ; in) ; d).

10. The probability of a reliable event is equal

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

Option 3.

If in this experience no two events can occur at the same time, then such events are called

a) incomplete;

b) impossible;

c) equivalent;

d) joint.

A combination of inconspicuous events such that as a result of experience should occur at least one of them is called

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

c) a holistic event system; d) not a holistic event system.

Working events BUT 1 and BUT 2

a) event occurs BUT 1 Event BUT 2 not happening;

b) event occurs BUT 2 Event BUT 1 not happening;

c) Events BUT 1 and BUT 2 happen simultaneously.

In the part of 100 parts 3 defects. What is the likelihood that the detail taken is defective?

but)
; b) ; in)
;
.

The sum of the probability of events forming the full system is equal

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

The probability of the impossible event is equal

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

BUT and IN Calculated by formula

a) p (a + c) \u003d p (a) + p (c); b) p (a + c) \u003d p (a) + p (c) - p (a c);

c) p (a + c) \u003d p (a) + p (c) + p (А c); d) p (a + c) \u003d p (a c) - p (a) + p (c).

On the shelf in random order, 10 textbooks are placed. Of these, 1 in mathematics, 2 in chemistry, 3 in biology and 4 in geography. Student raped 1 textbook. What is the probability that it will either in mathematics or in chemistry?

but) ; b); in) ; d) .

a) incomplete;

b) independent;

c) impossible;

d) dependent.

Two boxes are pencils equal size and forms. In the first box: 5 red, 2 blue and 1 black pencil. In the second box: 3 red, 1 blue and 2 yellow. Ruadach remove one pencil from each box. What is the likelihood that both pencils will be blue?

but) ; b) ; in) ; d) .

Option 2.

If an event occurs in this experience, it is called

a) joint;

b) real;

c) reliable;

d) impossible.

If the appearance of one of the events does not exclude the emergence of another in the same test, then such events are called

a) joint;

b) incomplete;

c) dependent;

d) independent.

If the occurrence of an event in does not have any influence on the likelihood of an event A, and vice versa, the occurrence of an event and does not have any influence on the likelihood of an event in, the events A and B are called

a) incomplete;

b) independent;

c) impossible;

d) dependent.

Sum of events BUT 1 and BUT 2 called an event that is carried out in the case when

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

b) Events BUT 1 and BUT 2 do not occur;

c) Events BUT 1 and BUT 2 happen simultaneously.

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) ; d).

The probability of the sum of two inconsistent events BUT and IN Calculated by formula

a) p (a + c) \u003d p (a) + p (c); b) p (a + c) \u003d p (a c) - p (a) + p (c);

c) p (a + c) \u003d p (a) + p (c) + p (А c); d) p (a + c) \u003d p (a) + p (c) - p (a c).

In the first box 2 white and 5 black balls. In the second box 2 white and 3 black bowls. From each box, the border was taken out 1 ball. What is the likelihood that both balls turn out to be black?

but) ; b); in) ; d).

The task

Option demo

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

b)

d)

e)

2. , - Event probabilities ,, 0 "style \u003d" margin-left: 55.05pt; Border-Collapse: Collapse; Border: None "\u003e

3. The probabilities of events and https://pandia.ru/text/78/195/images/image012_30.gif "width \u003d" 105 "height \u003d" 28 src \u003d "\u003e. Gif" width \u003d "55" height \u003d "24"\u003e there is:

a) 1.25 b) 0.3886 c) 0.25 g) 0,8614

e) no correct answer

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

Section 2. The probabilities of combining and crossing events, the conditional probability, the formula of the full probability and the Bayes.

The task: Select the correct answer and mark the corresponding letter in the table.

Option demo

1. Throw at the same time two playing bones. What is the probability that the amount of points dropped no more than 6?

but) ; b); in) ; d);

e) no correct answer

2. Each letter of the word "craft" is written on a separate card, then the cards are mixed. Take out three cards at random. What is the probability of getting the word "forest"?

but) ; b); in) ; d);

e) no correct answer

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

a) https://pandia.ru/text/78/195/images/image024_14.gif "width \u003d" 17 height \u003d 53 "height \u003d" 53 "\u003e; c); d); e) no correct answer

Test at the rate of probability theory and mathematical statistics.

Section 3. Discrete random variables and their numerical characteristics.

The task: Select the correct answer and mark the corresponding letter in the table.

Option demo

1 . Discrete random variables x and y are given by their laws

distributions

Random value z \u003d x + y. Find probability

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

2. X, Y, Z - independent discrete random variables. X value is distributed by binomial law with parameters n \u003d 20 and p \u003d 0.1. The value of Y is distributed according to a geometrical law with a parameter P \u003d 0.4. The value Z is distributed according to the law of Poisson with parameter \u003d 2. Find the dispersion of the random variable U \u003d 3x + 4Y-2Z

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

3. Two-dimensional random vector (x, y) set by the distribution law

Event, event . What is the likelihood of events a + in?

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

Test at the rate of probability theory and mathematical statistics.

Section 4. Continuous random variables and their numerical characteristics.

The task: Select the correct answer and mark the corresponding letter in the table.

Option demo

1. Independent continuous random variables X and Y are uniformly distributed on segments: x on https://pandia.ru/text/78/195/images/image032_6.gif "width \u003d" 32 "height \u003d" 23 "\u003e.

Random value Z \u003d 3X + 3Y +2. Find D (Z)

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

2 ..gif "width \u003d" 97 "height \u003d" 23 "\u003e

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

3. The continuous random value X is given by its probability density https://pandia.ru/text/78/195/images/image036_7.gif "width \u003d" 99 "height \u003d" 23 src \u003d "\u003e.

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

4. Random value x is distributed normally with parameters 8 and 3. Find

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

Test at the rate of probability theory and mathematical statistics.

Section 5. Introduction to Mathematical Statistics.

The task: Select the correct answer and mark the corresponding letter in the table.

Option demo

1. The following ratings are offered mathematical expectation https://pandia.ru/text/78/195/images/image041_6.gif "width \u003d" 98 "height \u003d" 22 "\u003e:

A) https://pandia.ru/text/78/195/images/image043_5.gif "width \u003d" 205 "height \u003d" 40 "\u003e

C) https://pandia.ru/text/78/195/images/image045_4.gif "width \u003d" 205 "height \u003d" 40 "\u003e

E) 0 "style \u003d" margin-left: 69.2pt; Border-Collapse: Collapse; Border: None "\u003e

2. The dispersion of each measurement in the previous task is. Then the most effective from the incredible estimates obtained in the first task will be

3. Based on the results of independent observations of the random variable of X, submitting to the Poisson law, to build an estimate of the unknown parameter 425 "STYLE \u003d" Width: 318.65pt; Margin-Left: 154.25pt; Border-Collapse: Collapse; Border: None "\u003e

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

4. Half-width 90% of the confidence interval built to assess the unknown mathematical expectation of a normally distributed random variable x for the sample volume n \u003d 120, selective middle https://pandia.ru/text/78/195/images/image052_3.gif "width \u003d" 19 "height \u003d" 16 "\u003e \u003d 5, there is

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

Check Matrix - Test Demo

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

Basic concepts on the topic:

1. Test, elemental outcome, test outcome, event.

2. A reliable event, an impossible event, a random event.

3. Joint events, incomplete events, equivalent events, equilibrium events, the only possible events.

4. Full group of events, opposite events.

5. Elementary event, composite event.

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

1. In the task, two target shots are produced. Find the likelihood that the target will be amazed once "the test is:

1) * produced two targets;

2) the target will be amazed once;

3) The target will be amazed twice.

2. Throw a coin. Event: A - "emblem will fall out." Having - "digits will fall" is:

1) random;

2) reliable;

3) impossible;

4) * Opposite.

3. Throw up dice. We denote the events: A - "Failure of 6 points", in - "Finding 4 points", D - "Loss of 2 points", C - "Favoring an even points." Then the event is equal

1)
;

2)
;

3)*
;

4)
.

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

1)*
;

2)
;

3)
;

4)
.

5. From the letters of the word "task" at random is selected one letter. Event - "Chosen Letter K" is

1) random;

2) reliable;

3) * impossible;

4) the opposite.

6. One letter is selected from the letters of the word "world". Event - "Chosen Letter M" is

1) * random;

2) reliable;

3) impossible.

7. Event - "From the urn containing only white balls, a white ball is removed" is

1) random;

2) * reliable;

3) impossible.

8. Two student pass the exam. Events: A - "Exam will hand over the first student", in - "Exam will hand over the second student" are

1) incomplete;

2) reliable;

3) impossible;

4) * joint.

9. Events are called unconditional if

4) * The occurrence of one excludes the possibility of the appearance of another.

10. Events are called the only possible if

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

2) when carrying out a complex of conditions, each of them has an equal opportunity to step;

3) * At the test, at least one of them will be completed;

Topic 2. Classical probability definition

Basic concepts on the topic:

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

2. Exodus favored by an event.

3. Geometric probability definition.

4. Relative event frequency.

5. Statistical definition of probability.

6. Properties of probability.

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

The use of all these concepts on practical examples.

Exemplary test tasks offered in this topic:

1. Events are called equivalence if

1) they are incomplete;

2) * When implementing a complex of conditions, each of them has an equal opportunity to step;

3) when testing will necessarily come at least one of them;

4) the offensive one excludes the possibility of the appearance of another.

2. Test - "Throw two coins." Event - "At least the coat of arms will fall on one of the coins." The number of elementary outcomes conducive to this event is:

4) four.

3. Test - "Throw two coins." Event - "The coat of arms will fall on one of the coins." The number of all elementary, equal, unique, possible, incomplete outcomes is:

4) * Four.

4. In the urn 12 balls, nothing different than color. Among these bowls are 5 black and 7 whites. Event - "Randomly remove a white ball". For this event, the number of favored outcomes is:

5. In the urn 12 balls, nothing excellent. Among these bowls are 5 black and 7 whites. Event - "Randomly remove a white ball". For this event, the number of all outcomes is:

6. The likelihood of an event takes any value from the gap:

3)
;

4)
;

5)*
.

7. The subscriber has forgotten the last two digits of the phone number and, knowing, only that they are different, scored them to make them. How many ways he can do it?

1);

2)*;

OPTION 1

1. In a random experiment throw two playing bones. Find the likelihood that 5 points fall in the amount. The result rounds to the hundredths.

2. In a random experiment, a symmetric coin is thrown three times. Find the chance that the eagle falls exactly twice.

3. In an average of 1,400 garden pumps arrived on sale, 7 leak. Find the likelihood that one randomly selected to control the pump does not leak.

4. The contest of the performers is held at 3 days. A total of 50 performances are stated - one from each country. On the first day, 34 performances, the rest are equally divided between the remaining days. The order of speeches is determined by the draw. What is the probability that the speech of the representative of Russia will take place on the third day of the competition?

5. In a taxi company in the presence of 50 passenger cars; 27 of them are black with yellow inscriptions on board, the rest are yellow with black inscriptions. Find the likelrough to a random call will arrive a yellow-colored machine with black inscriptions.

6. In the rock festival, groups are one from each of the stated countries. The order of performance is determined by the lot. What is the likelihood that a group from Germany will perform after a group from France and after the group from Russia? The result rounds to the hundredths.

7. What is the likelihood that randomly selected natural number From 41 to 56 divided into 2?

8. In the collection of tickets in mathematics, only 20 tickets, in 11 of them there is a question on logarithm. Find the likelihood that a schoolboy will get a question about logarithm in randomly selected on the exam.

9. The figure shows a labyrinth. The spider crashes into a labyrinth at the "Login" point. Expand and crawl the spider can not. On each branching, the spider chooses the path for which there is not yet plenty. Considering the choice of the further path random, determine with which probability of the spider will come to the output.

10. To enroll in the Institute for the specialty "Translator", the applicant must score at least 79 points on each of the three items - mathematics, Russian language and foreign language. To enroll on the specialty "Customs", you need to score at least 79 points for each of the three items - mathematics, Russian language and social studies.

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

Option 2.

1. In the store three sellers. Each of them is busy with a client with a probability of 0.3. Find the likelrough that at a random point of time all three sellers are occupied at the same time (consider that customers come independently from each other).

2. In a random experiment, a symmetric coin is thrown three times. Find the likelihood that the outcome of the RWP will come (the rush drops all three times).

3. The factory produces bags. On average, 200 high-quality bags account for four bags with hidden defects. Find the likelihood that the purchased bag will be high-quality. The result rounds to the hundredths.

4. The contest of the performers is held at 3 days. A total of 55 performances are stated - one from each country. On the first day, 33 performances, the rest are divided between the remaining days. The order of speeches is determined by the draw. What is the probability that the speech of the representative of Russia will take place on the third day of the competition?

5. On the phone keypad 10 digits, from 0 to 9. What is the likelihood that a randomly pressed digit will be less than 4?

6. Biathlonist 9 times shoots targets. The probability of getting into the target at one shot is 0.8. Find the chance that the biathlonist was first 3 times in the target, and the last six missed. The result rounds to the hundredths.

7. Two factories produce identical glass for automotive 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 likelihood that the glass bought in the store will be defective.

8. In the collection of tickets for chemistry only 25 tickets, in 6 of them there is a question on hydrocarbons. Find the likelihood that a schoolboy will get a question about hydrocarbons in randomly selected on the exam.

9. To enroll in the Institute for the specialty "Translator", the applicant must score at least 69 points on each of the three items - mathematics, Russian language and foreign language. To enroll on the specialty "Management", you need to score at least 69 points for each of the three items - mathematics, Russian language and social studies.

The probability that 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 social studies - 0.6.

Find the likelihood that T. will be able to enroll on one of the two mentioned specialties.

10. The figure shows a labyrinth. The spider crashes into a labyrinth at the "Login" point. Expand and crawl the spider can not. On each branching, the spider chooses the path for which there is not yet plenty. Considering the choice of the further path random, determine with which probability of the spider will come to the output.

Option 3.

1. 60 athletes participate in the gymnastics championship: 14 from Hungary, 25 from Romania, the rest - from Bulgaria. The order in which gymnasts perform is determined by the lot. Find the likelihood that the athlete protruding the first will be from Bulgaria.

2. Automatic line manufactures batteries. The likelihood that the finished battery is faulty is 0.02. Before packing, each battery is undergoing the control system. The probability that the system takes the faulty battery is 0.97. The likelihood that the error system takes the serviceable battery is 0.02. Find the likelihood that the battery selected from the package will be rejected.

3. In order to enroll in the Institute for the specialty "International Relations", the applicant must score at least 68 points on each of the three items - mathematics, Russian language and foreign language. To enroll on the specialty "Sociology", you need to score at least 68 points on each of the three items - mathematics, Russian language and social studies.

The likelihood that Eating V. will receive at least 68 points in mathematics, equal to 0.7, in the Russian language - 0.6, in a foreign language - 0.6 and social studies - 0.7.

Find the likelihood that V. will be able to enroll on one of the two mentioned specialties.

4. The figure shows a labyrinth. The spider crashes into a labyrinth at the "Login" point. Expand and crawl the spider can not. On each branching, the spider chooses the path for which there is not yet plenty. Considering the choice of the further path random, determine with which probability of the spider will come to the output.

5. What is the likelihood that randomly chosen natural number from 52 to 67 is divided into 4?

6. On the geometry exam, a single question comes from the list of examination issues. The likelihood that this is a question on the topic "inscribed circle" is 0.1. The likelihood that this is a question on the topic "trigonometry" is 0.35. Questions that simultaneously refer to these two topics, no. Find the likelihood that the student's exam will get a question on one of these two topics.

7. Seva, Glory, Anya, Andrey, Misha, Igor, Nadia and Karina threw lot - who to start the game. Find the likelihood that the boy should start the game.

8. 5 scientists from Spain, 4 from Denmark and 7 from Holland arrived at the seminar. The procedure of the reports is determined by the draw. Find the likelihood that the twelfth is a report of a scientist from Denmark.

9. In the collection of tickets on philosophy, only 25 tickets, 8 of them meet the question of Pythagora. Find the likelihood that a schoolboy will not get a question in a ticket for a ticket for a ticket on Pythagora.

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

Option 4.

1. In the rock festival, groups are among one from each of the stated countries. The order of performance is determined by the lot. What is the likelihood that the group from the United States will perform after a group of Vietnam and after the group from Sweden? The result rounds to the hundredths.

2. The likelihood that on the history of the history student T. will correctly solve more than 8 tasks, equal to 0.58. The likelihood that T. will correctly solve more than 7 tasks, equal to 0.64. Find the likelihood that T. correctly solves exactly 8 tasks.

3. The factory produces bags. On average, 60 high-quality bags account for six bags with hidden defects. Find the likelihood that the purchased bag will be high-quality. The result rounds to the hundredths.

4. In the pocket of Sasha had four candy - "Bear", "takeoff", "protemat" and "grill", as well as the keys to the apartment. Having taken out the keys, Sasha accidentally dropped one candy out of his pocket. Find the likelihood that the "takeoff" candy was lost.

5. The figure shows a labyrinth. The spider crashes into a labyrinth at the "Login" point. Expand and crawl the spider can not. On each branching, the spider chooses the path for which there is not yet plenty. Considering the choice of the further path random, determine with which probability of the spider will come to the output.

6. Three playing bones are thrown in a random experiment. Find the likelihood that 15 points fall out. The result rounds to the hundredths.

7. Biathlonist shoots 10 times on targets. The probability of hitting the target at one shot is 0.7. Find the chance that the biathlonist was first 7 times in the target, and the last three missed. The result rounds to the hundredths.

8. 5 scientists from Switzerland, 7 from Poland and 2 from the UK arrived at the seminar. The procedure of the reports is determined by the draw. Find the likelihood that the report of a scientist from Poland will be the thirteenth.

9. To enroll in the Institute for the specialty " International law", Applicant must score at least 68 points on each of the three items - mathematics, Russian and foreign language. To enroll on the specialty "Sociology", you need to score at least 68 points on each of the three items - mathematics, Russian language and social studies.

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

Find the likelihood that B. will be able to enroll on one of the two mentioned specialties.

10. B. mall Two identical machines sell coffee. The likelihood that by the end of the day in the machine will end coffee, equal to 0.25. The likelihood that coffee will end in both machines is 0.14. Find the likelihood that by the end of the day the coffee will remain in both machines.