Brought to date in open jar 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 There are 9 red balls and 3 blue ones in the basket. 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 be blue?

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

Decision. Now we calculate the probability of choosing a blue ball.
Event A: "the chosen ball turned out to be blue"
Total number of all possible outcomes: 9+3=12 (number of all balls we could draw)
Number of outcomes favorable 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 for the same problem the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. The number of favorable outcomes: 9. The desired probability: 9/12=3/4=0.75

The probability of any event always lies between 0 and 1.
Sometimes in everyday speech (but not in probability theory!) 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,
In this case, the probability is zero for events that cannot happen - improbable. For example, in our example, this would be the probability of drawing a green ball from the basket. (The number of favorable outcomes is 0, P(A)=0/12=0 if counted according to the formula)
Probability 1 has events that will absolutely definitely happen, without options. For example, the probability that "the chosen 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 that illustrates the definition of probability. All similar USE tasks according to probability theory are solved by applying this formula.
Instead of red and blue balls, there can be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on a certain 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 USE probability theory, 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 an elementary outcome, and then apply the same formula.

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

What is the elementary outcome here? - Assigning a professor's report to one of all possible serial numbers for a speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report 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, matching was considered in terms of which of the places a particular person could take. You can approach the same situation from the other side: which of the people with what probability could get to a particular place (prototypes , , , ):

Example 3 5 Germans, 8 Frenchmen and 3 Estonians participate in the draw. What is the probability that the first (/second/seventh/last - it doesn't 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 - the French. 8 people.
Desired probability: 8/16=1/2=0.5
Answer: 0.5

The prototype is slightly different. There are tasks about coins () and dice () that are somewhat more creative. Solutions to these problems can be found on the prototype pages.

Here are some examples of coin tossing or dice tossing.

Example 4 When we toss a coin, what is the probability of getting tails?
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 that it will come up heads both times?
The main thing is to determine which elementary outcomes we will consider when tossing two coins. After tossing two coins, one of the following results can occur:
1) PP - both times it came up tails
2) PO - first time tails, second time heads
3) OP - the first time heads, the second time tails
4) OO - heads up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one is favorable, 1.
Probability: 1/4=0.25
Answer: 0.25

What is the probability that two tosses of a coin will land on tails?
The number of elementary outcomes is the same, 4. Favorable outcomes are the second and third, 2.
Probability of getting one tail: 2/4=0.5

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

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

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

Example 6 We throw a dice. What is the probability of getting an even number?

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

Example 7 Throw two dice. What is the probability that the total rolls 10? (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 a total of 10 to fall out?
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. So, for cubes, options are possible:
(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)
In total, 3 options. Desired probability: 3/36=1/12=0.08
Answer: 0.08

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

Exercise

Demo option

1. and are 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

d) there is no correct answer

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

Section 2. Probabilities of combining and crossing 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 sum of the rolled points is not greater than 6?

a) ; b) ; in) ; G) ;

d) there is no correct answer

2. Each letter of the word "CRAFT" is written on a separate card, then the cards are mixed. We take out three cards at random. What is the probability of getting the word "WOOD"?

a) ; b) ; in) ; G) ;

d) there is no correct answer

3. Among the second-year students, 50% never missed classes, 40% missed classes 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 rest - 10% received the highest score. The student received the highest score on the exam. Find the probability that he missed 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) 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 set by their own laws

distribution

Random variable Z = X+Y. Find Probability

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

2. X, Y, Z are independent discrete random variables. The X value is distributed according to the binomial law with parameters n=20 and p=0.1. The Y value is distributed according to the geometric law with the parameter p=0.4. The value of Z is distributed according to the Poisson law with the parameter =2. Find the variance of a random variable U= 3X+4Y-2Z

a) 16.4 b) 68.2; c) 97.3; d) 84.2; d) 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 event A+B?

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

Test on 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 uniformly 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; d) there is no correct answer

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

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

Test on 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 mathematical expectation estimates 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">

C) 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. The variance of each measurement in the previous problem is . Then the most efficient of the unbiased estimates obtained in the first problem is the estimate

3. Based on the results of independent observations of a random variable X obeying the Poisson law, construct an estimate of the unknown parameter by the method of moments 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; d) there is no correct answer

4. FWHM 90% confidence interval, built to estimate the unknown mathematical expectation of a normally distributed random variable X for 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; d) there is no correct answer

Validation Matrix - test demo

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

Option 1.

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

a) cannot happen

b) either happens or it doesn't;

c) will definitely happen.

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

a) equivalent;

b) joint;

c) simultaneous;

d) identical.

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

a) opposite;

b) incompatible;

c) impossible;

d) equivalent.

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

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

The probability of a certain event is equal to

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

Probability of product of two dependent events BUT and AT calculated by the formula

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

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

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

a) ; b) ; in) ; G) .

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

a) ; b) ; in) ; G) .

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

a) ; b) ; in) ; G) .

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

a) b) ; in) ; G) .

Option 4.

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

a) impossible;

b) incompatible;

c) optional;

d) unreliable.

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

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

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

a) elementary;

b) incompatible;

c) impossible;

d) reliable.

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

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

a) ; b); in) ; G) .

Probability of product of two independent events BUT and AT calculated by the formula

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

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

There are 20 people in the class. Of these, 5 are excellent students, 9 are good students, 3 have triples and 3 have deuces. What is the probability that a randomly selected student is either a good student or an excellent student?

a) ; b) ; in) ; G) .

9. The first box contains 2 white and 3 black balls. The second box contains 4 white and 5 black balls. One ball is drawn at random from each box. What is the probability that both balls are white?

a) ; b) ; in) ; G) .

10. The probability of a certain event is equal to

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

Option 3.

If in a given experiment no two of the events can occur simultaneously, then such events are called

a) incompatible;

b) impossible;

c) equivalent;

d) joint.

A set of incompatible events such that at least one of them must occur as a result of the experiment is called

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

c) an integral system of events; d) not an integral system of events.

The product of events BUT 1 and BUT 2

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

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

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

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

a)
; b) ; in)
;
.

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

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

The probability of an impossible event is

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

BUT and AT calculated by the formula

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

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

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

a) ; b) ; in) ; G) .

a) incompatible;

b) independent;

c) impossible;

d) dependent.

Two boxes contain pencils of the same size and shape. In the first box: 5 red, 2 blue and 1 black pencil. In the second box: 3 red, 1 blue and 2 yellow. One pencil is drawn at random from each box. What is the probability that both pencils are blue?

a) ; b) ; in) ; G) .

Option 2.

If an event necessarily occurs in a given experience, then it is called

a) joint;

b) real;

c) reliable;

d) impossible.

If the occurrence of one of the events does not exclude the occurrence of another in the same trial, then such events are called

a) joint;

b) incompatible;

c) dependent;

d) independent.

If the occurrence of event B does not have any effect on the probability of occurrence of event A, and vice versa, the occurrence of event A does not have any effect on the probability of occurrence of event B, then events A and B are called

a) incompatible;

b) independent;

c) impossible;

d) dependent.

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

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

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

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

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

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

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

a) ; b) ; in) ; G) .

Probability of the sum of two incompatible events BUT and AT calculated by the formula

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

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

The first box contains 2 white and 5 black balls. The second box contains 2 white and 3 black balls. One ball is drawn at random from each box. What is the probability that both balls are black?

a) ; b) ; in) ; G) .

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

a) medical statistics

b) probability theory

c) medical demographics

d) higher mathematics

Correct answer: b

2. THE POSSIBILITY OF IMPLEMENTING ANY EVENT IS:

a) experiment

b) scheme of cases

c) regularity

d) probability

The correct answer is g

3. EXPERIMENT IS:

a) the process of accumulation of empirical knowledge

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

c) study covering the entire population of observation units

d) mathematical modeling of reality processes

Correct answer b

4. OUTCOME IN PROBABILITY THEORY IS UNDERSTANDING:

a) an uncertain result of the experiment

b) a certain result of the experiment

c) the dynamics of the probabilistic process

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

Correct answer b

5. SAMPLE SPACE IN PROBABILITY THEORY IS:

a) the structure of the phenomenon

b) all possible outcomes of the experiment

c) the ratio between two independent sets

d) the ratio between two dependent populations

Correct answer b

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

a) frequency of occurrence

b) probability

c) a phenomenon

d) an event

The correct answer is g

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

a) random

b) equiprobable

c) equivalent

d) selective

Correct answer b

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

a) needed

b) expected

c) reliable

d) priority

Correct answer in

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

a) unnecessary

b) unexpected

c) impossible

d) non-priority

Correct answer in

10. PROBABILITY OF A RANDOM EVENT:

a) greater than zero and less than one

b) more than one

c) less than zero

d) represented by whole numbers

Correct answer a

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

a) will always appear

b) will appear in 90% of experiments

c) will appear in 95% of experiments

d) will appear in 99% of experiments

Correct answer a

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

The correct answer is g

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

a) credible

b) incompatible

c) random

d) probable

Correct answer b

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

a) equal

b) joint

c) equally likely

d) incompatible

Correct answer in

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

a) random

b) equally possible

c) selective

d) total

Correct answer a

16. IF WE KNOW THE NUMBER OF POSSIBLE OUTCOMES OF A SOME EVENT AND THE TOTAL NUMBER OF OUTCOMES IN THE SAMPLE SPACE, THEN WE CAN CALCULATE:

a) conditional probability

b) classical probability

c) empirical probability

d) subjective probability

Correct answer b

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

a) conditional probability

b) classical probability

c) empirical probability

d) subjective probability

Correct answer in

18. BASED ON YOUR PERSONAL OBSERVATIONS, YOU DO:

a) objective probability

b) classical probability

c) empirical probability

d) subjective probability

The correct answer is g

19. THE SUM OF TWO EVENTS BUT And AT THE EVENT IS CALLED:

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

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

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

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

Correct answer in

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

a) the joint occurrence of events A and B

b) consecutive appearance of events A and B

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

d) the occurrence of either event A or event B

Correct answer a

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

a) independent

b) ungrouped

c) remote

d) heterogeneous

Correct answer a

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

a) homogeneous

b) grouped

c) one-time

d) dependent

The correct answer is g

23. PROBABILITY ADDITION THEOREM:

a) the probability of the sum of two joint events is equal to the sum of the probabilities of these events

b) the probability of the successive occurrence of two joint events is equal to the sum of the probabilities of these events

c) the probability of the sum of two incompatible events is equal to the sum of the probabilities of these events

d) the probability of non-occurrence of two incompatible events is equal to the sum of the probabilities of these events

Correct answer in

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

a) empirical probability tends to classical

b) the empirical probability moves away from the classical

c) subjective probability exceeds the classical one

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

Correct answer a

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

a) probability multiplication theorem

b) probability addition theorem

c) Bayes' theorem

d) Bernoulli's theorem

Correct answer a

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

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

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

Correct answer in

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

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

b) the probability of producing independent events is equal to the product of the probabilities of these events

c) if event A does not depend on event B, then event B does not depend on event A

d) the probability of the product of dependent events is equal to the product of the probabilities of these events

Correct answer b

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

a) a priori

b) a posteriori

c) preliminary

d) initial

Correct answer a

29. PROBABILITIES REVISED AFTER ADDITIONAL INFORMATION IS REVIEWED ARE CALLED

a) a priori

b) a posteriori

c) preliminary

d) final

Correct answer b

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

a) Bernoulli

b) Bayesian

c) Chebyshev

d) Poisson

Correct answer b

BUT) !

B)

b)

G) P(A)=

The order is not important when using

A) placements

B) permutations

B) combinations

D) permutations and placements

A) 12 131415=32760

B) 13 1415=2730

AT 12 1314=2184

D) 14 15=210

Combination of n elements by m-This

A) the number of subsets containingm elements

B) the number of place changes by an element of a given set

C) the number of ways to choosem elements from nc order

D) the number of ways to choosem elements from nregardless of order

How many ways are there to seat the quartet from the fable of the same name by I.A. Krylov?

A) 24

B) 4

AT 8

D) 6

In how many ways can one headman and one fizorg be chosen from a group of 30 people?

A) 30

B) 870

B) 435

D) 30!

BUT)

B)

AT)

G)

BUT)

B) ( m-2)(m-1)m

B) (m-1)m

G) ( m-2)(m-1)

In how many ways can a group of 30 send 5 people to run the college run?

A) 17100720

B) 142506

B) 120

D) 30!

The eight students shook hands. How many handshakes were there?

A) 40320

B) 28

C) 16

D) 64

How many ways can you choose 3 books out of 9 given?

BUT)

B)

C) R 9

D) 3P 9

There are 5 red and 3 white roses in a vase. In how many ways can 4 flowers be taken?

BUT)

B)

AT)

G)

There are 8 red and 3 white roses in a vase. In how many ways can you take 2 red and 1 white roses?

BUT)

B)

AT)

G)

A) 110

B) 108

AT 12

D) 9

There are 38 branches in the mailbox. In how many ways can 35 identical cards be placed in a box so that each box contains at most one card?

BUT)

B) 35!

AT)

D) 38!

How many different permutations can be formed from the word "elephant"?

A) 6

B) 4

C) 24

D) 8

In how many ways can two items be selected from a box containing 10 items?

A) 10!

B) 90

C) 45

D) 100

How many different two-digit numbers can be formed from the numbers 1,2,3,4?

A) 16

B) 24

AT 12

D) 6

3 vouchers are allocated for 5 employees. In how many ways can they be distributed if all vouchers are different?

A) 10

B) 60

B) 125

D) 243

A) (6;+ )

B) (- ;6)

B) (0; + )

D) (0;6)

BUT)

B)

AT)

G)

A) 4

B) 3

IN 2

D) 5

Write down the formula the phrase "the number of combinations ofnelements of 3 are 5 times less than the number of combinations ofn+2 elements of 4 »

BUT)

B)

AT)

G)

In how many ways can 28 students be seated in a lecture hall?

A) 2880

B) 5600

C) 28!

D) 7200

In how many ways can 25 workers form teams of 5 people each?

A) 25!

B)

AT)

D) 125

There are 26 students in the group. In how many ways can 2 people be assigned to duty so that one of them is the leader?

BUT)

B)

C) 24!

D) 52

A) 6

B) 5

AT)

D) 15

How many five-digit numbers can be formed from the digits 1,2,3,4,5 without repetitions?

A) 24

B) 6

B) 120

D) 115

How many five-digit numbers can be formed from the digits 1,2,3,4,5 so that 3 and 4 are side by side?

A) 120

B) 6

B) 117

D) 48

The Scientific Society consists of 25 members. It is necessary to choose the president of the society, the vice-president, the scientific secretary and the treasurer. In how many ways can this choice be made if each member of the society must hold only one position?

A) 303600

B) 25!

B) 506

D) 6375600

BUT) ( n-4)(n-5)

B) ( n-2)(n-1)n

AT)

G)

A) -2

B) -3

IN 2

D) 5

In how many ways can 8 rooks be placed on a chessboard so that they cannot attack each other?

A) 70

B) 1680

C) 64

D) 40320

BUT)

B) (2 m-1)

AT) 2m

D) (2 m-2)!

BUT) ( n-5)!

B)

AT)

G) n(n-1)(n-2)

A) 6

B) 4

AT 5

D) 3

A) -1

B) 6

B) 27

D)-22

A) 1

B) 0

IN 3

D) 4

A) 9

B) 0.5

C) 1.5

D) 0.3

The combination is calculated by the formula

BUT) !

B)

B) P(A)=

G)

Accommodations are calculated using the formula

BUT) P(A)=

B)

b)

G) !

Permutations from n elements is

A) the choice of elements from the set "n»

B) the number of elements in the set "n»

C) a subset of a set ofn elements

D) the established order in the set "n»

Placements are applied in the problem if

A) there is a choice of elements from the set, taking into account the order

B) there is a choice of elements from a set without regard to order

C) it is necessary to carry out a permutation in the set

D) if all selected elements are the same

An urn contains 6 white and 5 black balls. In how many ways can 2 white and 3 black balls be drawn from it?

BUT)

B)

AT)

G)

Among 100 lottery tickets, 45 are winning. In how many ways can one win out of three purchased tickets?

A) 45

B)

AT)

G)

Answers to test number 1

Answers to test number 2

Test #2

"Fundamentals of Probability Theory"

It's called a random event.

A) such an outcome of the experiment, in which the expected result may or may not occur

B) such an outcome of the experiment, which is already known in advance

C) an outcome of the experiment that cannot be determined in advance

D) such an outcome of the experiment, which, while maintaining the conditions of the experiment, is constantly repeated

conjunction "and" means

A) addition of probabilities of events

B) multiplying the probabilities of events

D) division of probabilities of events

conjunction "or" means

A) division of probabilities of events

B) addition of probabilities of events

C) the difference in the probabilities of events

D) multiplication of the probabilities of events

Events in which the occurrence of one precludes the occurrence of the other are called

A) incompatible

B) independent

B) dependent

D) joint

The complete group of events is formed by

A) a set of independent events, if as a result of single tests one of these events necessarily occurs

B) a set of independent events, if as a result of single tests all these events will necessarily occur

C) a set of incompatible events, if as a result of single tests one of these events necessarily occurs

D) a set of incompatible events, if as a result of single tests all these events will necessarily occur

The opposite are called

A) two independent, forming a complete group, events

B) two independent events

B) two incompatible events

D) two incompatible, forming a complete group, events

Two events are called independent

A) which as a result of the test will necessarily occur

B) which as a result of the test never occur together

C) in which the outcome of one of them does not depend on the outcome of the other event

D) in which the outcome of one of them is completely dependent on the outcome of another event

An event that is sure to occur as a result of the test

A) impossible

B) accurate

B) authentic

D) random

An event that will never happen as a result of the test

A) impossible

B) accurate

B) authentic

D) random

Highest value probabilities are

A) 100%

B) 1

B) infinity

D) 0

The sum of the probabilities of opposite events is equal to

A) 0

B) 100%

IN 1

D) 1

The phrase "at least one" means

A) only one element

B) not a single element

D) one, two or no more elements

The classical definition of probability

A) the probability of an event is the ratio of the number of outcomes that favor the occurrence of an event to the number of all incompatible, unique and equally possible outcomes that form a complete group of events.

B) Probability is a measure of the possibility of an event occurring in a particular test

C) Probability is the ratio of the number of trials in which an event occurred to the number of trials in which the event could or may not have occurred.

D) Each random event A from the field of events is assigned a non-negative number P(A), called probability.

Probability is a measure of the possibility of an event occurring in a particular test.

This is the definition of probability

A) classic

B) geometric

B) axiomatic

D) statistical

Probability is the ratio of the number of trials in which an event occurred to the number of trials in which the event might or might not have occurred. This is the definition of probability

A) classic

B) geometric

B) axiomatic

D) statistical

The conditional probability is calculated by the formula

A) P (A / B) \u003d

B) P (A + B) \u003d P (A) + P (B) -P (AB)

C) P (AB) \u003d P (A) P (B)

D) P (A + B) \u003d P (A) + P (B)

This formula P (A + B) \u003d P (A) + P (B) -P (AB) is used for two

A) incompatible events

B) joint events

B) dependent events

D) independent events

For which two events does the concept of conditional probability apply?

A) impossible

B) reliable

B) joint

D) dependent

Total Probability Formula

A) R( H I /A)=

B) P(A)=P(A/ H 1 ) P(H 1 )+ P(A/ H 2 ) P(H 2 )+…+ Р(А/ H n ) P(H n )

AT) P n (m)=

D) P(A)=

B) Bayes' theorem

B) Bernoulli scheme

A) total probability formula

B) Bayes' theorem

B) Bernoulli scheme

D) classical definition of probability

Two dice are thrown. Find the probability that the sum of the rolled points is 6

A) P(A)=

B) P(A)=

B) P(A)=

D) P(A)=

Two dice are thrown. Find the probability that the sum of the rolled points is 11 and the difference is 5

A) P(A)=0

B) P(A)=2/36

C) P(A) = 1

D) P(A)=1/6

The device, which operates during the day, consists of three nodes, each of which, independently of the others, can fail during this time. Failure of any of the nodes disables the entire device. The probability of correct operation during the day of the first node is 0.9, the second - 0.85, the third - 0.95. What is the probability that the device will work during the day without fail?

A) P(A)=0.1 0.15 0.05=0.00075

B) P(A)=0.9 0.85 0.95=0.727

C) P(A)=0.1+0.85 0.95=0.91

D) P(A)=0.1 0.15 0.95=0.014

A two-digit number is conceived, the digits of which are different. Find the probability that a randomly named two-digit number will be equal to the intended number?

A) P(A)=0.1

B) P(A)=2/90

C) P (A) \u003d 1/100

D) P(A)=0.9

Two people shoot at a target with the same probability of hitting 0.8. What is the probability of hitting the target?

A) P(A)=0.8 0.8=0.64

B) P(A)=1-0.2 0.2=0.96

C) P(A)=0.8 0.2+0.2 0.2=0.2

D) P(A)=1-0.8=0.2

Two students are looking for the book they need. The probability that the first student finds the book is 0.6, and the second is 0.7. What is the probability that only one of the students will find the right book?

A) P(A)=1-0.6 0.7=0.58

B) P(A)=1-0.4 0.3=0.88

C) P(A)=0.6 0.3+0.7 0.4=0.46

D) P(A)=0.6 0.7+0.3 0.4=0.54

From a deck of 32 cards, two cards are taken at random one after the other. Find the probability that two kings are drawn?

A) P(A)=0.012

B) P (A) \u003d 0.125

C) P(A)=0.0625

D) P(A)=0.031

Three shooters independently shoot at a target. The probability of hitting the target for the first shooter is 0.75, for the second 0.8, for the third 0.9. Find the probability that at least one shooter hits the target?

A) P (A) \u003d 0.25 0.2 0.1 \u003d 0.005

B) P(A)=0.75 0.8 0.9=0.54

C) P(A)=1-0.25 0.2 0.1=0.995

D) P(A)=1-0.75 0.8 0.9=0.46

There are 10 identical parts in a box, marked with numbers from #1 to #10. Randomly take 6 parts. Find the probability that part number 5 will be among the extracted parts?

A) P (A) \u003d 5/10 \u003d 0.2

B) P(A)=

C) P (A) \u003d 1/10 \u003d 0.1

D) P(A)=

Find the probability that among 4 products taken at random, 3 will be defective if there are 10 defective products in a batch of 100 products.

A) P(A)=

B) P(A)=

B) P(A)=

D) P(A)=

There are 10 white and 8 scarlet roses in a vase. Two flowers are chosen at random. What is the probability of that. What are they different colors?

A) P(A)=

B) P(A)=

B) P(A)=

D) P(A) = 2/18

The probability of hitting the target with one shot is 1/8. What is the probability that out of 12 shots there will be no misses?

A) R 12 (12)=

B) R 12 (1)=

B) P(A)=

D) P(A)=

The goalkeeper parries an average of 30% of all penalty kicks. What is the probability that he will take 2 out of 4 balls?

A) R 4 (2)=

B) R 4 (2)=

C) R 4 (2)=

D) R 4 (2)=

There are 40 vaccinated rabbits and 10 controls in the nursery. 14 rabbits are checked in a row, the result is recorded and the rabbits are sent back. Determine the most likely number of appearances of the control rabbit.

A) 10

B) 14

C) 14

D) 14

Top grade products at the shoe factory account for 10% of all production. How many pairs of top quality boots can you hope to find among the 75 pairs that came from this factory to the store?

A)75

B) 75

B) 75

D) 75

A) Local Laplace formula

B) Laplace integral formula

B) Moivre-Laplace formula

D) Bernoulli scheme

When solving the problem “The probability of the appearance of defects in a series of parts is 2%. What is the probability that in a batch of 600 parts there will be 20 defective ones? more applicable

A) Bernoulli scheme

B) De Moivre-Laplace formula

B) local Laplace formula

When solving the problem “In each of 700 independent tests for marriage, the appearance of a standard light bulb occurs with a constant probability of 0.65. Find the probability that, under these conditions, the occurrence of a defective bulb will occur more often than in 230 trials, but less often than in 270 trials” is more applicable

A) Bernoulli scheme

B) De Moivre-Laplace formula

B) local Laplace formula

D) Laplace integral formula

When dialing a phone number, the subscriber forgot the number and dialed it at random. Find the probability that the desired number is dialed?

A) P(A)=1/9

B) P(A)=1/10

C) P(A)=1/99

D) P(A)=1/100

A dice is thrown. Find the probability of getting an even number of points?

A) P (A) \u003d 5/6

B) P(A)=1/6

C) P(A)=3/6

D) P(A)=1

There are 50 identical parts in a box, 5 of them are painted. One piece is drawn at random. Find the probability that the extracted part will be painted?

A) P(A)=0.1

B) P(A)=

B) P(A)=

D) P(A)=0.3

An urn contains 3 white and 9 black balls. Two balls are taken out of the urn at the same time. What is the probability that both balls are white?

A) P(A)=

B) P(A)=

C) P(A)=2/12

D) P(A)=

10 different books are placed at random on one shelf. Find the probability that 3 certain books will be placed side by side?

A) P(A)=

B) P(A)=

B) P (A) \u003d

D) P(A)=

Participants in the draw draw tokens with numbers from 1 to 100 from the box. Find the probability that the number of the first randomly drawn token does not contain the number 5?

A) P(A)=5/100

B) P(A)=1/100

B) P(A)=

D) P(A)=

Test #3

"Discrete Random Variables"

A quantity that, depending on the result of the experiment, can take on different numerical values, is called

A) random

B) discrete

B) continuous

D) probability

A discrete random variable is called

A) a value that, depending on the result of the experiment, can take on different numerical values

B) a value that changes from one test to another with a certain probability

C) a value that does not change during several tests

D) a value that, regardless of the result of the experiment, can take on different numerical values

Fashion is called

A) the average value of a discrete random variable

B) the sum of the products of the values ​​of a random variable by their probability

C) the mathematical expectation of the square of the deviation of a value from its mathematical expectation

D) the value of a discrete random variable, the probability of which is the greatest

The mean value of a discrete random variable is called

A) fashion

B) mathematical expectation

B) median

The sum of the products of the values ​​of a random variable and their probability is called

A) variance

B) mathematical expectation

B) fashion

D) standard deviation

Mathematical expectation of the squared deviation of a value from its mathematical expectation

A) fashion

B) median

B) standard deviation

D) dispersion

The formula by which the variance is calculated

BUT)

B) M (x 2) -M (x)

C) M (x 2) - (M (x)) 2

D) (M (x)) 2 -M (x 2)

The formula by which the mathematical expectation is calculated

BUT)

B) M (x 2) - (M (x)) 2

AT)

G)

For a given series of distribution of a discrete random variable, find the mathematical expectation

A) 1

B) 1.3

B) 0.5

D) 0.8

For a given series of distribution of a discrete random variable, find M(x 2 )

A) 1.5

B) 2.25

B) 2.9

D) 0.99

Find unknown probability

A) 0.65

B) 0.75

C) 0

D) 1

Find fashion

A) 0.03

B) 1.7

C) 0.28

D) 1.2

Find Median

A) 0.08

B) 1.2

AT 4

D) 0.28

Find Median

A) 1.2

B) 3.5

B) 0.25

D) 1.1

To find unknown value x if M(x)=1.1

A) 3

B) 1.1

B) 1.2

D) 0

The mathematical expectation of a constant value is