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. 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) necessary
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 OCCURING AND CANNOT DETERMINE THE NUMBER OF POSSIBLE OUTCOMES OF THE EVENT INTERESTING 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
Option number 1
- There are 14 defective bricks in a batch of 800 bricks. The boy chooses at random one brick from this batch and throws it from the eighth floor of the construction site. What is the probability that a thrown brick will be defective?
- The physics examination book for grade 11 consists of 75 tickets. In 12 of them there is a question about lasers. What is the probability that Step's student, choosing a ticket at random, will stumble upon a question about lasers?
- 3 athletes from Italy, 5 athletes from Germany and 4 from Russia compete at the 100m championship. The lane number for each athlete is determined by a draw. What is the probability that an athlete from Italy will be on the second lane?
- 1500 bottles of vodka were delivered to the store. It is known that 9 of them are overdue. Find the probability that an alcoholic who chooses one bottle at random will end up buying the expired one.
- There are 120 offices of various banks in the city. Grandma chooses one of these banks at random and opens a deposit of 100,000 rubles in it. It is known that during the crisis, 36 banks went bankrupt, and the depositors of these banks lost all their money. What is the probability that Granny will not lose her deposit?
- In one 12-hour shift, a worker produces 600 parts on a CNC machine. Due to a defect in the cutting tool, 9 defective parts were received on the machine. At the end of the working day, the workshop foreman takes one part at random and checks it. What is the probability that he will get exactly the defective part?
Test on the topic: "Probability theory in the tasks of the exam"
Option number 1
- At the Kievsky railway station in Moscow, there are 28 ticket windows, next to which 4,000 passengers are crowding, wishing to buy train tickets. According to statistics, 1680 of these passengers are inadequate. Find the probability that the cashier sitting behind the 17th window will encounter an inadequate passenger (taking into account that passengers choose the cashier at random).
- Russian Standard Bank holds a lottery for its customers - holders of Visa Classic and Visa Gold cards. 6 Opel Astra cars, 1 Porsche Cayenne car and 473 iPhone 4 phones will be raffled off. It is known that the manager Vasya issued a Visa Classic card and became the winner of the lottery. What is the probability that he will win an Opel Astra if the prize is chosen at random?
- In Vladivostok, a school was renovated and 1,200 new plastic windows were installed. An 11th grade student who did not want to take the USE in mathematics found 45 cobblestones on the lawn and started throwing them at the windows at random. In the end, he broke 45 windows. Find the probability that the window in the director's office is not broken.
- A batch of 9,000 counterfeit Chinese-made chips has arrived at an American military factory. These microcircuits are installed in electronic sights for the M-16 rifle. 8766 ICs are known to be defective in this batch and scopes with such ICs will not function correctly. Find the probability that a randomly selected electronic sight works correctly.
- Granny keeps 2,400 jars of cucumbers in the attic of her country house. It is known that 870 of them have long been rotten. When the granddaughter came to the granny, she gave him one jar from her collection, choosing it at random. What is the probability that the granddaughter received a jar of rotten cucumbers?
- A team of 7 migrant construction workers offers apartment renovation services. During the summer season, they completed 360 orders, and in 234 cases they did not remove construction waste from the entrance. Public utilities choose one apartment at random and check the quality of the repair work. Find the probability that utility workers will not stumble upon building debris when checking.
Answers:
Var#1 | ||||||
answer | 0,0175 | 0,16 | 0,25 | 0,006 | 0,015 |
|
Var #2 | ||||||
answer | 0,42 | 0,0125 | 0,9625 | 0,026 | 0,3625 | 0,35 |
Tests by discipline"Probability Theory and math statistics»
Option 1
What is the mathematical expectation of the random variable X?
a) 1; b) 2; at 4; d) 2.5; e) 3.5.
| ||||||||||||
| q J | ||||||||||||
What is the mathematical expectation of a random variable 
?
a) 0.5; b) 0; c) 0.3; d) 2.2; e) 3.
| Measurement number | ||||||||
| x i |
Determine the unbiased estimate of the variance.
a) 48.5; b) 341.7; c) 12.9; d) 63.42; e) 221.1.
Option 2
a) Bernoulli's formula; b) Laplace's local theorem; c) Laplace's integral theorem; d) Poisson's formula.
The mathematical expectation of a random variable X distributed according to the binomial law is:
a) npq; b) np; c) nq; d) pq.
The Laplace function has the following property: Ф(0)=0.
a) true; b) incorrect.
The correlation coefficient characterizes the degree of tightness of the linear relationship between random variables
a) true; b) incorrect.
The distribution matrix of a system of two discrete random variables (X, Y) is given by the table
| y i x i | |||
What is the variance of the random variable Y.
a) 2; b) 5; c) 3.5; d) 2.56; e) 2.2.
| ||||||||||||
| q J | ||||||||||||
What is the variance of a random variable ?
a) 0.9; b) 0.3; c) 1.15; d) 5.6; e) 0.21.
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 given 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 over geometric law with 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. Half-width of 90% confidence interval constructed 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 | ||||||
TEST #1
Topic: Types of random events, classical definition of probability,
elements of combinatorics.
You are offered 5 test items on the topic types of random events, the classical definition of probability, elements of combinatorics. Among the suggested answers only one is true.
| Exercise | Suggested answers |
|
| If the occurrence of an event BUT affects the probability value of event B, then about events BUT and AT they say they... | joint; incompatible; dependent; independent. |
|
| There are 5 flags on the garland different color. You can count the number of possible combinations of them using: | the formula for the number of placements; formula for the number of permutations; formula for the number of combinations; |
|
| Among the 100 banknotes received at the cash desk, 8 are counterfeit. The cashier randomly takes out one bill. The probability that this banknote will be accepted at the bank is equal to: | ||
| The 25 seater bus includes 4 passengers. They can take any seat on the bus. The number of ways these people can be placed on the bus is calculated by the formula: | number of permutations; number of combinations; number of placements; |
|
| The dice is thrown once. Dropping the number "4" on the top face is: | certain event; impossible event; random event. |
TEST #2
Topic: Theorems of addition and multiplication of probabilities.
You are offered 5 test tasks on the topic of the theorem of addition and multiplication of probabilities. Among the suggested answers only one is true.
| Exercise | Suggested answers |
|
| An event consisting in the fact that either an event will occur BUT, or an event AT can be designated: |
A-B; |
|
| Formula P(A+B) = P(A) + P(B), corresponds to the probability addition theorem: | dependent events; independent events; joint events; incompatible events. |
|
| The miss probability for a torpedo boat is . The boat fired 6 shots. The probability that the boat hit the target all 6 times is equal to: | ||
| Probability of joint occurrence of events BUT and AT stand for: | |
|
| The problem is given: in the first box - 5 white and 3 red balls, in the second - 3 white and 10 red balls. One ball was drawn at random from each box. Determine the probability that both balls are the same color. To solve the problem use: | The theorem of multiplication of probabilities of incompatible events and the theorem of addition of probabilities of independent events. The theorem of addition of probabilities of incompatible events; The theorem of multiplication of probabilities of independent events and the theorem of addition of probabilities of incompatible events; The theorem of multiplication of probabilities of dependent events; |
TEST #3
Topic: Random independent trials according to the Bernoulli scheme.
You are offered 5 test tasks on the topic of random independent tests according to the Bernoulli scheme. Among the suggested answers only one is true.
| Suggested answers |
||
| Given the task: The probability that there is a typo on the page of a student's abstract is 0.03. The abstract consists of 8 pages. Determine the probability that exactly 5 of them are misspelled. | Bernoulli formula; Laplace's local theorem; Laplace's integral theorem; Poisson formula. |
|
| The family plans to have 5 children. If we assume the probability of having a boy is 0.515, then the most probable number of girls in the family is equal to: | ||
| There is a group of 500 people. Find the probability that two people have a birthday on New Year. Assume that the probability of being born on a fixed day is . To solve this problem, use: | Bernoulli formula; Laplace's local theorem; Laplace's integral theorem; Poisson formula. |
|
| To determine the probability that in 300 trials an event BUT happens at least 40 times, if the probability A in each trial is constant and equal to 0.15, use: | Bernoulli's formula and the addition theorem for the probabilities of incompatible events; Laplace's local theorem; Laplace's integral theorem; Poisson's formula, the addition theorem for the probabilities of incompatible events, the property of the probabilities of opposite events. |
|
| The problem is given: it is known that in some area in September there are 18 rainy days. What is the probability that out of seven days randomly taken in this month, two days will be rainy? To solve this problem, use: | Bernoulli formula; Laplace's local theorem; Laplace's integral theorem; Poisson formula. |
TEST #4
Topic: One-dimensional random variables.
You are offered 5 test tasks on the topic of one-dimensional random variables, their ways of setting and numerical characteristics. Among the suggested answers only one is true.