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 " maths 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; at 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; at 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) .
Presented to date in the open bank of USE problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is a 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.
Solution. 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 favorable outcomes for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P(A)=3/12=1/4=0.25
Answer: 0.25
Let us calculate 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 problems in probability theory are solved using 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 at one toss of a coin we have 2 possible outcomes, then for two tosses of results there will be 2 2=2 2 =4 (as in example 5), for three tosses 2 2 2=2 3 =8, for four: 2·2·2·2=2 4 =16, … for N throws of possible outcomes there will be 2·2·...·2=2 N .
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.
1 option
1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=m=100
2. They threw a dice. What is the probability of getting an even number of points?
Answer:
1 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - all parts are defective.
Answer:
- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and at least one boiler are running.
Answer:
5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 5.
Answer:
6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: all the boys will be in the same subgroup?
7. A coin was flipped 3 times. What is the probability that heads will come up 3 times.
Answers:
8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is white.
Answers:
9. Choose the correct answer:
Answers:
10. Choose the correct answer: Total probability formula
11. Find P (AB) if
Answers:
12. Find if P(A) = 0.2
13. Events A and B are incompatible. Find P(A + B) if P(A) = P(B) = 0.3
14. Find P (A + B) if P (A) \u003d P (B) \u003d 0.3 P (AB) \u003d 0.1
15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 10, m = 2
16. The most probable number of occurrences of an event when repeating tests is found by the formula:
17. The sum of the products of each DSV value and the corresponding probability is called.
p = 0.9; n = 10
p = 0.9; n = 10
22. . The binomial law of distribution of DSV is given. Find P(x
23. Find the appropriate formula: M(x) =?
Answers:
Find .
Answers:
Answers:
27. A random variable has a uniform distribution if
Answers:
Answers:
Answer: a) b)
c) d)
30. In formula
Answers:
Test on the subject "Probability Theory and Mathematical Statistics"
Option 2
1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=1000; m=100
Answer: a) 0.75 b) 1 c) 0.5 d) 0.1
2. They threw a dice. What is the probability of getting more than four
Answer:
3. There are 20 standard parts and 7 defective parts in a box. Pulled out three parts. Event A 1 – 1st part is defective, A 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - all parts are standard.
Answer:
4. Let A - the machine works, B- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and at least two boilers are running.
Answer:
5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 8.
Answer:
6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 2 young men will be in one subgroup, and 4 in another?
Answers a) 8 b) 168 c) 840 d) 56
7. A coin was flipped 3 times. What is the probability that heads will come up once.
Answers:
8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is blue.
Answers:
9. Choose the correct answer:
Answers:
10. Choose the correct answer: Bernoulli formula
11. Find P (AB) if
Answers:
12. Find if P(A) = 0.8
Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6
13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.25 P(B) = 0.45
Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6
14. Find P (A + B) if P (A) \u003d 0.2 P (B) \u003d 0.8 P (AB) \u003d 0.1
Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7
15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 20, m = 3
Answers: a) b) 0.2 c) 0.25 d) 0.15
16. Local Moivre-Laplace theorem
17. The mathematical expectation of the square of the difference between the random variable X and its mathematical expectation is called:
Answers: a) the variance of a random variable b) the mathematical expectation of the DSV
C) standard deviation d) DSV distribution law
18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).
p = 0.8; n = 9
Answers: a) 8.4 b) 6 c) 7.2 d) 9
19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).
p = 0.8; n = 9
Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9
20. The binomial law of distribution of DSV is given. Find M(x).
Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8
21. The binomial law of distribution of DSV is given. Find D(x).
Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84
22. The binomial law of distribution of DSV is given. Find P (x > 2).
Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792
23. Find the appropriate formula: D (x) \u003d?
Answers:
24. The law of distribution of DSV is given. Find M(x).
Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9
25. The law of distribution of DSV is set. Find.
Answers:
Answers:
27. A random variable has a normal distribution if
Answers:
28. Find the differential distribution function f(x)if
Answers:
29. Find the integral distribution function F(x) if
Answer: a) b)
c) d)
30. In formula
Answers:
Test on the subject "Probability Theory and Mathematical Statistics"
3 option
1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=500 m=255
Answer: a) 0.75 b) 1 c) 0.5 d) 0.1
2. They threw a dice. What is the probability of getting less than five
Answer:
3. There are 20 standard parts and 7 defective parts in a box. Pulled out three parts. Event A 1 – 1st part is defective, A 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - at least one part is defective.
Answer:
4. Let A be a machine, B- the -th boiler is working ( =1,2,3). Record event: unit is running The machine-boiler unit is running if the machine and all boilers are running.
Answer:
5. An n-volume collected works were placed on the shelf in random order. What is the probability that there are 100 booksyat in ascending order of volume numbers if n = 10.
Answer:
6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 3 young men will be in one subgroup, and 3 in another?
Answers a) 8 b) 168 c) 840 d) 56
7. A coin was flipped 3 times. What is the probability that heads will come up at least once.
Answers:
8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is yellow.
Answers:
9. Choose the correct answer:
Answers:
10. Choose the correct answer: Bayss formula
11. Find P (AB) if
Answers:
12. Find if P(A) = 0.5
Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6
13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.7 P(B) = 0.1
Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6
14. Find P (A + B) if P (A) \u003d 0.5 P (B) \u003d 0.2 P (AB) \u003d 0.1
Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7
15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 40, m = 10
Answers: a) b) 0.2 c) 0.25 d) 0.15
16. Laplace integral theorem
17. The square root of the dispersion of a random variable is called:
Answers: a) the variance of a random variable b) the mathematical expectation of the DSV
C) standard deviation d) DSV distribution law
18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).
p = 0.7; n = 12
Answers: a) 8.4 b) 6 c) 7.2 d) 9
19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).
p = 0.7; n = 12
Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9
20. The binomial law of distribution of DSV is given. Find M(x).
Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8
21. The binomial law of distribution of DSV is given. Find D(x).
Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84
22. The binomial law of distribution of DSV is given. Find P(0
Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792
(x) = ?
Answers:
24. The law of distribution of DSV is given. Find M(x).
Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9
25. The law of distribution of DSV is set. Find
Answers:
Answers:
27. A random variable has an exponential distribution if
Answers:
28. Find the differential distribution function f(x)if
Answers:
29. Find the integral distribution function F(x) if
Answer: a) b)
c) d)
30. In formula
Answers:
Test on the subject "Probability Theory and Mathematical Statistics"
4 option
1. The experiment was performed n times, the event A happened m times. Find the frequency of occurrence of event A: n=400 m=300
Answer: a) 0.75 b) 1 c) 0.5 d) 0.1
2. They threw a dice. What is the probability of getting less than six
Answer:
3. There are 20 standard parts and 7 defective parts in a box. Pulled out three parts. Event A 1 – 1st part is defective, A 2 – 2nd part is defective, A 3 – The 3rd part is defective. Record event: B - one part is defective and two are standard.
Answer:
4. Let A be a machine, B- the -th boiler is working ( =1,2,3). Log an event: the unit is running the machine-boiler unit is running if the machine is running; 1st boiler and at least one of the other two boilers.
Answer:
5. An n-volume collected works were placed on the shelf in random order. What is the probability that the books are in ascending order of volume numbers if n = 7.
Answer:
6. There are 8 girls and 6 boys in the group. They were divided into two equal subgroups. How many outcomes favor the event: 5 young men will be in one subgroup, and 1 in another?
Answers a) 8 b) 168 c) 840 d) 56
7. A coin was flipped 3 times. What is the probability that heads will come up more than once.
Answers:
8. There are 25 balls in a box, of which 10 are white, 7 are blue, 3 are yellow, and 5 are blue. Find the probability that a randomly drawn ball is blue.
Answers:
9. Choose the correct answer:
Answers:
10. Choose the correct answer: The formula for the product of the probabilities of dependent events
11. Find P (AB) if
Answers:
12. Find if P(A) = 0.4
Answers: a) 0.5 b) 0.8 c) 0.2 d) 0.6
13. Events A and B are incompatible. Find P(A + B) if P(A) = 0.6 P(B) = 0.3
Answers: a) 0.9 b) 0.8 c) 0.7 d) 0.6
14. Find P (A + B) if P (A) \u003d 0.6 P (B) \u003d 0.4 P (AB) \u003d 0.4
Answers: a) 0.5 b) 0.6 c) 0.9 d) 0.7
15. The experience was made n times. Event A happened m times. Find the frequency of occurrence of event A: n = 60, m = 10
Answers: a) b) 0.2 c) 0.25 d) 0.15
16. Bernoulli's theorem
17. A correspondence that establishes a relationship between the possible values of a random variable and their probabilities is called:
Answers: a) the variance of a random variable b) the mathematical expectation of the DSV
C) standard deviation d) DSV distribution law
18. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find M(x).
p = 0.6; n = 10
Answers: a) 8.4 b) 6 c) 7.2 d) 9
19. The probability of failure-free operation of one cell of the milking machine is p. X is the number of non-failure cells of the milking machine during the milking of n cows. Find D(x).
p = 0.6; n = 10
Answers: a) 2.52 b) 3.6 c) 1.44 d) 0.9
20. The binomial law of distribution of DSV is given. Find M(x).
Answers: a) 2.8 b) 1.2 c) 2.4 d) 0.8
21. The binomial law of distribution of DSV is given. Find D(x).
Answers: a) 0.96 b) 0.64 c) 0.36 d) 0.84
22. . The binomial law of distribution of DSV is given. Find P(1
Answers: a) 0.0272 b) 0.0272 c) 0.3398 d) 0.1792
23. Find the corresponding formula:
Answers:
24. The law of distribution of DSV is given. Find M(x).
Answer: a) 3.8 b) 4.2 c) 0.7 d) 1.9
25. The law of distribution of DSV is set. Find
Answers:
Answers:
27. A random variable has a binomial distribution if
Answers:
28. Find the differential distribution function f(x)if
Answers:
29. Find the integral distribution function F(x) if
Answer: a) b)
c) d)
30. In formula
Answers:
TEST #1
Topic: Types of random events, classical definition of probability,
elements of combinatorics.
You are offered 5 test tasks 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. |
|
| On the garland hang 5 flags of different colors. 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's Eve. 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.
OPTION 1
1. In a random experiment, two dice are thrown. Find the probability of getting 5 points in total. Round the result to the nearest hundredth.
2. In a random experiment, a symmetrical coin is thrown three times. Find the probability that heads come up exactly twice.
3. On average, out of 1,400 garden pumps sold, 7 leak. Find the probability that one randomly selected pump does not leak.
4. The competition of performers is held in 3 days. There are 50 entries in total, one from each country. There are 34 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place on the third day of the competition?
5. The taxi company has 50 cars; 27 of them are black with yellow inscriptions on the sides, the rest are yellow with black inscriptions. Find the probability that a yellow car with black inscriptions will arrive at a random call.
6. Groups perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a group from Germany will perform after a group from France and after a group from Russia? Round the result to the nearest hundredth.
7. What is the probability that a randomly chosen natural number from 41 to 56 is divisible by 2?
8. There are only 20 tickets in the collection of tickets in mathematics, 11 of them contain a question on logarithms. Find the probability that a student will get a logarithm question in a ticket randomly selected in the exam.
9. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
10. To enter the institute for the specialty "Translator", the applicant must score at least 79 points on the Unified State Examination in each of the three subjects - mathematics, Russian and a foreign language. To enter the specialty "Customs", you need to score at least 79 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant B. will receive at least 79 points in mathematics is 0.9, in Russian - 0.7, in a foreign language - 0.8 and in social studies - 0.9.
OPTION 2
1. There are three sellers in the store. Each of them is busy with a client with a probability of 0.3. Find the probability that at a random moment of time all three sellers are busy at the same time (assume that customers enter independently of each other).
2. In a random experiment, a symmetrical coin is tossed three times. Find the probability that the outcome of the RPP will come (all three times it comes up tails).
3. The factory produces bags. On average, for every 200 quality bags, there are four bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to the nearest hundredth.
4. The competition of performers is held in 3 days. There are 55 entries in total, one from each country. There are 33 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place on the third day of the competition?
5. There are 10 digits on the telephone keypad, from 0 to 9. What is the probability that a randomly pressed number will be less than 4?
6. Biathlete shoots at targets 9 times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hit the targets the first 3 times and missed the last 6. Round the result to the nearest hundredth.
7. Two factories produce the same glass for car headlights. The first factory produces 30 of these glasses, the second - 70. The first factory produces 4 defective glasses, and the second - 1. Find the probability that a glass randomly bought in a store will be defective.
8. There are only 25 tickets in the collection of chemistry tickets, 6 of them contain a question on hydrocarbons. Find the probability that a student will get a question on hydrocarbons in a ticket randomly selected in the exam.
9. In order to enter the institute for the specialty "Translator", the applicant must score at least 69 points on the Unified State Examination in each of the three subjects - mathematics, Russian and a foreign language. To enter the specialty "Management", you need to score at least 69 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant T. will receive at least 69 points in mathematics is 0.6, in Russian - 0.6, in a foreign language - 0.5 and in social studies - 0.6.
Find the probability that T. will be able to enter one of the two specialties mentioned.
10. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
OPTION 3
1. 60 athletes participate in the gymnastics championship: 14 from Hungary, 25 from Romania, the rest from Bulgaria. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete who competes first is from Bulgaria.
2. Automatic production line for batteries. The probability that a finished battery is defective is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a bad battery is 0.97. The probability that the system will mistakenly reject a good battery is 0.02. Find the probability that a randomly selected battery will be rejected.
3. To enter the Institute for the specialty "International Relations", the applicant must score at least 68 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Sociology", you need to score at least 68 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant V. will receive at least 68 points in mathematics is 0.7, in Russian - 0.6, in a foreign language - 0.6 and in social studies - 0.7.
Find the probability that B. will be able to enter one of the two specialties mentioned.
4. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
5. What is the probability that a randomly chosen natural number from 52 to 67 is divisible by 4?
6. On the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.1. The probability that this is a trigonometry question is 0.35. There are no questions related to these two topics at the same time. Find the probability that the student will get a question on one of these two topics on the exam.
7. Seva, Slava, Anya, Andrey, Misha, Igor, Nadya and Karina cast lots for who to start the game. Find the probability that a boy will start the game.
8. 5 scientists from Spain, 4 from Denmark and 7 from Holland came to the seminar. The order of reports is determined by a draw. Find the probability that the report of a scientist from Denmark will be the twelfth.
9. There are only 25 tickets in the collection of tickets on philosophy, 8 of them contain a question on Pythagoras. Find the probability that a student will not get a question on Pythagoras in a ticket randomly selected at the exam.
10. There are two payment machines in the store. Each of them can be faulty with a probability of 0.09, regardless of the other automaton. Find the probability that at least one automaton is serviceable.
OPTION 4
1. Groups perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the likelihood that a band from the USA will perform after a band from Vietnam and after a band from Sweden? Round the result to the nearest hundredth.
2. The probability that student T. correctly solves more than 8 problems on the history test is 0.58. The probability that T. correctly solves more than 7 problems is 0.64. Find the probability that T. correctly solves exactly 8 problems.
3. The factory produces bags. On average, for every 60 quality bags, there are six bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to the nearest hundredth.
4. Sasha had four sweets in his pocket - “Mishka”, “Vzlyotnaya”, “Squirrel” and “Roasting”, as well as the keys to the apartment. Taking out the keys, Sasha accidentally dropped one candy from his pocket. Find the probability that the take-off candy is lost.
5. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
6. In a random experiment, three dice are thrown. Find the probability of getting 15 points in total. Round the result to the nearest hundredth.
7. Biathlete shoots at targets 10 times. The probability of hitting the target with one shot is 0.7. Find the probability that the biathlete hit the targets the first 7 times and missed the last 3. Round the result to the nearest hundredth.
8. 5 scientists from Switzerland, 7 from Poland and 2 from Great Britain came to the seminar. The order of reports is determined by a draw. Find the probability that the thirteenth is the report of a scientist from Poland.
9. To enter the institute for the specialty "International Law", the applicant must score at least 68 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Sociology", you need to score at least 68 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant B. will receive at least 68 points in mathematics is 0.6, in Russian - 0.8, in a foreign language - 0.5 and in social studies - 0.7.
Find the probability that B. will be able to enter one of the two specialties mentioned.
10. There are two identical coffee machines in the mall. The probability that the machine will run out of coffee by the end of the day is 0.25. The probability that both machines will run out of coffee is 0.14. Find the probability that by the end of the day there will be coffee left in both vending machines.
