Problems for independent solution. Problems for independent solution Problems for independent work
4.1. Each of the four incompatible events can occur with probabilities of 0.012, 0.010, 0.006 and 0.002, respectively. Determine the probability that at least one of these events will occur as a result of the experiment.
(Answer: p = 0.03)
4.2. The shooter fires one shot at a target consisting of a central circle and two concentric rings. The probabilities of hitting the circle and the ring are 0.20, 0.15 and 0.10, respectively. Determine the probability of missing the target.
(Answer: p = 0.55)
4.3. Two identical coins of radius r are located inside a circle of radius R into which a point is thrown at random. Determine the probability that this point will fall on one of the coins if the coins do not overlap.
(Answer: p = )
4.4. What is the probability of drawing a figure of any suit or a card of spades from a deck of 52 cards (the figure is called a jack, queen or king)?
(Answer: p = 
)
4.5. The box contains 10 coins of 20 kopecks, 5 coins of 15 kopecks. and 2 coins of 10 kopecks. Six coins are taken at random. What is the probability that the total will be no more than one ruble?
(Answer: p = 
)
4.6. Two urns contain balls that differ only in color, and in the first urn there are 5 white balls, 11 black and 8 red, and in the second there are 10, 8 and 6, respectively. One ball is drawn at random from both urns. What is the probability that both balls are the same color?
(Answer: p = 0.323)
4.7. The game between A and B is played under the following conditions: as a result of the first move, which A always makes, he can win with probability 0.3; if A does not win with the first move, then B makes the move and can win with probability 0.5; if as a result of this move B does not win, then A makes a second move, which can lead to his win with probability 0.4. Determine the probabilities of winning for A and B.
(Answer: 
= 0,44, = 0,35)
4.8. The probability for a given athlete to improve his previous result in one attempt is equal to p. Determine the probability that an athlete will improve his result at a competition if two attempts are allowed.
(Answer: p(A) = )
4.9. From an urn containing n balls with numbers from 1 to n, two balls are drawn sequentially, with the first ball being returned if its number is not equal to one. Determine the probability that ball number 2 will be drawn the second time.
(Answer: p = )
4.10. Player A plays alternately with players B and C, having a probability of winning in each game of 0.25, and stops playing after the first loss or after two games played with each player. Determine the probabilities of winning B and C.
4.11. Two people take turns tossing a coin. The one who gets the coat of arms first wins. Determine the probabilities of winning for each player.
(Answer: 
)
4.12. The probability of getting a point without losing serve when two equal volleyball teams play is equal to half. Determine the probability of getting one point for the serving team.
(Answer: p = )
4.13. Two shooters take turns shooting at the target until the first hit is made. The probability of a hit for the first shooter is 0.2, and for the second it is 0.3. Find the probability that the first shooter will fire more shots than the second.
(Answer: p = 0.455)
4.14. Two players play until victory, and for this the first needs to win m games, and the second n games. The probability of winning each game by the first player is p, and the second q=1-p. Determine the probability of the first player winning the entire game.
, Criminal Procedure Code of the Russian Federation from 18.1.rtf , Fundamentals of the legislation of the Russian Federation on health protection , ECtHR. Legal mechanism for filing an individual complaint and legal.
Lesson 4. Theorem of addition of probabilities.
14.1. Brief theoretical part
The probability of the sum of two events is determined by the formulaP( A+IN) = P( A)+P( B) - R( AB),
which generalizes to the sum of any number of events
For incompatible events, the probability of the sum of events is equal to the sum of the probabilities of these events, i.e.
24.2. Test
In what case are events A and B called incompatible or incompatible?
b) When at least one of these events occurs during the test
c) When the joint occurrence of these events is impossible
d) When both of these events occur during the experiment
Specify events that are compatible.
b) The presence of the same student at the same time at a lecture in the classroom and in the cinema
c) The onset of spring according to the calendar and snowfall
d) Appearance on the dropped edge of each of the two dice three points and equality of points on the sides of both dice rolled in an odd number
e) Showing a football match on one television channel and a news broadcast on another
The theorem for adding the probabilities of incompatible events is formulated as follows:
b) The probability of the occurrence of one of two incompatible events is equal to the sum of the probabilities of these events
c) The probability of the occurrence of one of two incompatible events is equal to the difference in the probabilities of the occurrence of these events
The theorem for adding the probabilities of joint events is formulated as follows:
b) The probability of the occurrence of at least one of two joint events is equal to the sum of the probabilities of these events without the probability of their joint occurrence
c) The probability of the occurrence of at least one of two joint events is equal to the sum of the probabilities of these events and the probability of their joint occurrence
The addition theorem of probabilities is generalized to the sum of any number of events and the probability of the sum of events in general form is calculated by the formula:
If events are incompatible, then the probability of the sum of these events is equal to:
b)
V)
34.3. Solving typical problems
Example 4.1. Determine the probability that that a batch of one hundred products, among which five are defective, will be accepted upon testing a randomly selected half of the entire batch, if the conditions of acceptance allow no more than one out of fifty defective products.Solution.
WITH, consisting in the fact that a batch of one hundred products, including five defective ones, will be accepted when testing a randomly selected half of the entire batch.
Let us denote by A an event consisting in the fact that during testing not a single defective product was received, and through IN- the event that only one defective product is received.
Since C=A+B, then the desired probability P(C) = P( A+B).
Events A And IN incompatible. Therefore P(C) = P( A)+ P( B).
Out of 100 products, 50 can be selected in different ways. Of the 95 non-defective products, 50 can be selected using methods.
Therefore P( A)=.
Similar to P( B)= .
P(C) = P( A)+ P( B)=+==0,181.
Example 4.2.
Electrical circuit between points M And N compiled according to the diagram shown in Fig. 5.
Failure over time T various circuit elements - independent events, having the following probabilities (Table 1).
Table 1
Element K 1
K 2
L 1
L 2
L 3
Probability0,60,50,40,70,9 Determine the probability of a circuit break for a specified period of time.
Solution.
Let us introduce the event WITH, consisting in the fact that within a specified period of time there will be a break in the circuit.
Let us denote by A j (j= 1.2) event consisting of failure of an element TO j, through A- failure of at least one element TO j, and through IN- failure of all three elements A i (i=1, 2, 3).
Then the desired probability
R( WITH) = P( A + IN) = P( A) + P( IN) - R( A)R( B).
R( A) = P( A 1 ) + P( A 2 ) - R( A 1 )R( A 2 ) = 0,8,
R( IN) = P( L 1 )R( L 2 ) R( L 3 ) = 0,252,
That.
Example 4.3.
The urn contains n white, m black and l red balls, which are drawn at random one at a time:
a) without return;
b) with return after each extraction.
In both cases, determine the probability that the white ball will be drawn before the black one.
Solution.
Let R 1 - the probability that the white ball will be drawn before the black one, A R 11 - the probability that the black ball will be drawn before the white one.
Probability R 1 is the sum of the probabilities of drawing a white ball immediately, after drawing one red, two red, etc. Thus, we can write in the case when the balls are not returned,
and when the balls return
To obtain probabilities R 11
in the previous formulas you need to make a replacement n on m, A m on n. It follows that in both cases R 1
:R 11
= n:m. Since, in addition, R 1
+R 11
= 1, then the required probability when removing balls without returning is also equal.
Example 4.4.
Someone wrote n letters, sealed them in envelopes, and then randomly wrote different addresses on each of them. Determine the probability that at least one of the envelopes has the correct address written on it.
Solution.
Let the event A k is that on k- the envelope contains the correct address ( k= l, 2,..., n).
The desired probability.
Events A k joint; for any different k, j, i, ... the following equalities hold:
Using the formula for the probability of the sum n events, we get
At large n.
44.4. Tasks for independent work
4.1. Each of the four incompatible events can occur with probabilities of 0.012, 0.010, 0.006 and 0.002, respectively. Determine the probability that at least one of these events will occur as a result of the experiment.(Answer:
p = 0.03)
4.2. The shooter fires one shot at a target consisting of a central circle and two concentric rings. The probabilities of hitting the circle and the ring are 0.20, 0.15 and 0.10, respectively. Determine the probability of missing the target.
(Answer:
p = 0.55)
4.3. Two identical radius coins r located inside a circle of radius R, into which a point is thrown at random. Determine the probability that this point will fall on one of the coins if the coins do not overlap.
(Answer:
p =)
4.4. What is the probability of drawing a figure of any suit or a card of spades from a deck of 52 cards (the figure is called a jack, queen or king)?
(Answer:
p =)
4.5. The box contains 10 coins of 20 kopecks, 5 coins of 15 kopecks. and 2 coins of 10 kopecks. Six coins are taken at random. What is the probability that the total will be no more than one ruble?
(Answer:
p =)
4.6. Two urns contain balls that differ only in color, and in the first urn there are 5 white balls, 11 black and 8 red, and in the second there are 10, 8 and 6, respectively. One ball is drawn at random from both urns. What is the probability that both balls are the same color?
(Answer:
p = 0.323)
4.7. Game between A And B is carried out under the following conditions: in the result of the first move who always does A, he can win with probability 0.3; if the first move A doesn't win, then makes a move IN and can win with probability 0.5; if as a result of this move IN doesn't win, then A makes a second move, which can lead to his winning with probability 0.4. Determine the probabilities of winning for A and for IN.
(Answer:
= 0,44, = 0,35)
4.8. The probability for a given athlete to improve his previous result in one attempt is R. Determine the probability that an athlete will improve his result at a competition if two attempts are allowed.
(Answer:
p(A) =)
4.9. From an urn containing n balls with numbers from 1 to n, two balls are drawn sequentially, with the first ball being returned if its number is not one. Determine the probability that ball number 2 will be drawn the second time.
(Answer:
p =)
4.10. Player A takes turns playing with players IN And WITH, having a probability of winning in each game of 0.25, and stops the game after the first loss or after two games played with each player. Determine the probabilities of winning IN And WITH.
(Answer:
)
4.11. Two people take turns tossing a coin. The one who gets the coat of arms first wins. Determine the probabilities of winning for each player.
(Answer:
)
4.12. The probability of getting a point without losing serve when two equal volleyball teams play is equal to half. Determine the probability of getting one point for the serving team.
(Answer:
p =)
4.13. Two shooters take turns shooting at the target until the first hit is made. The probability of a hit for the first shooter is 0.2, and for the second it is 0.3. Find the probability that the first shooter will fire more shots than the second.
(Answer:
p = 0.455)
4.14. Two players play until victory, and for this the first one must win T parties, and the second P parties. The probability of the first player winning each game is R, and the second q=1-R. Determine the probability of the first player winning the entire game.
(Answer: p(A) =)
1. The first box contains 2 white and 10 black balls; The second box contains 8 white and 4 black balls. A ball was taken from each box. What is the probability that both balls are white?
2. The first box contains 2 white and 10 black balls; The second box contains 8 white and 4 black balls. A ball was taken from each box. What is the probability that one ball is white and the other is black?
3. There are 6 white and 8 black balls in a box. Two balls are taken out of the box (without returning the removed ball to the box). Find the probability that both balls are white.
4. Three shooters shoot at the target independently of each other. The probability of hitting the target for the first shooter is 0.75, for the second – 0.8, for the third – 0.9. Determine the probability that all three shooters will hit the target at the same time; at least one shooter will hit the target.
5. There are 9 white and 1 black balls in the urn. Three balls were taken out at once. What is the probability that all the balls are white?
6. Fire three shots at one target. The probability of hitting each shot is 0.5. Find the probability that these shots will result in only one hit.
7. Two shooters, for whom the probabilities of hitting the target are 0.7 and 0.8, respectively, fire one shot each. Determine the probability of at least one hit on the target.
8. The probability that the part produced on the first machine will be first-class is 0.7. When the same part is manufactured on the second machine, this probability is 0.8. The first machine produced two parts, the second three. Find the probability that all parts are first-class.
9. The operation of the device stopped due to the failure of one lamp out of five . Finding this lamp is done by replacing each lamp with a new one in turn. Determine the likelihood that you will have to check 2 lamps, if the probability of failure of each lamp is p = 0.2 .
10. On the site AB For a motorcyclist-racer there are 12 obstacles, the probability of stopping at each of them is 0.1. The probability that from point IN to the final destination WITH the motorcyclist will travel without stopping, equal to 0.7. Determine the probability that on the site AC there won't be a single stop.
11. There are 4 traffic lights on the car’s path. The probability of stopping at the first two is 0.3, and at the next two is 0.4. What is the probability of driving through traffic lights without stopping?
12. There are 3 traffic lights on the car’s path. The probability of stopping at the first two is 0.4, and at the third is 0.5. What is the probability of passing traffic lights with one stop?
13. Two Internet servers are exposed to the risk of a virus attack per day with a probability of 0.3. What is the probability that there was not a single attack on them in 2 days?
14. The probability of hitting the target with one shot for a given shooter is 2/3. If a hit is recorded on the first shot, then the shooter is entitled to the second. If he hits again at the second time, he shoots a third time. What is the probability of hitting with three shots?
15. Game between A And IN is carried out under the following conditions: as a result of the first move, which always makes A, he can win with probability 0.3; if the first move A doesn't win, then makes a move IN and can win with probability 0.5; if as a result of this move IN doesn't win, then A makes a second move, which can lead to his winning with probability 0.4. Determine the probabilities of winning for A and for IN.
16. The probability for a given athlete to improve his previous result in one attempt is 0.2 . Determine the probability that an athlete will improve his result at a competition if two attempts are allowed.
17. Player A alternately plays two games with the players IN And WITH. Probabilities of winning the first game for IN And WITH equal to 0.1 and 0.2, respectively; probability of winning in the second game for IN is equal to 0.3, for WITH equal to 0.4. Determine the probability that: a) B will win first; b) will be the first to win WITH.
18. From an urn containing P balls with numbers from 1 to n, two balls are drawn sequentially, with the first one returned if its number is not equal to one. Determine the probability that ball number 2 will be drawn the second time.
19. Player A alternately plays with players B and C, having a probability of winning in each game of 0.25, and stops the game after the first win or after two games lost with either player. Determine the probabilities of winning B and C.
20. Two people take turns tossing a coin. The one who wins is the one. which the coat of arms will appear first. Determine the probabilities of winning for each player.
21. An urn contains 8 white and 6 black balls. Two players draw one ball in succession, returning the removed ball each time. The game continues until one of them gets the white ball. Determine the probability that the player starting the game will be the first to draw the white ball.
22. A courier was sent to collect documents from 4 archives. Probability of presence necessary documents in the first archive – 0.9; in II – 0.95; in III – 0.8; in IV – 0.6. Find the probability P of the absence of a document in only one archive.
23. Find the probability that two of three independently operating elements of a computing device will fail if the probability of failure of the first, second and third elements, respectively, are 0.3, 0.5, 0.4.
24. There are 8 white and 4 gray mice in a cage. Three mice are randomly selected for laboratory testing and not returned. Find the probability that all three mice are white.
25. There are 8 guinea pigs in a cage. Three of them suffer from a violation of the metabolism of mineral salts. Three animals are taken out in succession without returning. What is the probability that they are healthy?
26. The pond contains 12 crucian carp, 18 bream and 10 carp. Three fish were caught. Find the probability that you caught two carp and a crucian carp in succession.
27. There are 12 cows in the herd, 4 of which are Simmental breeds, the rest are Galstein-Friesian breeds. Three animals were selected for breeding work. Find the probability that all three of them are Simmental breeds.
28. At the hippodrome there are 10 bay horses, 3 dapple gray and 7 white. 2 horses were randomly selected for the race. What is the probability that there is no white horse among them?
29. The kennel contains 9 dogs, of which 3 are collies, 2 are boxers, the rest are Great Danes. Three dogs are randomly selected. What is the probability that at least one of them is a boxer?
30. The average offspring of animals is 4. The appearance of female and male individuals is equally probable. Find the probability that the offspring contains two males.
31. The bag contains seeds whose germination rate is 0.85. The probability that the plant will bloom is 0.9. What is the probability that a plant grown from a random seed will bloom?
32. The bag contains bean seeds, the germination rate of which is 0.9. The probability that the bean flowers will be red is 0.3. What is the probability that a plant from a randomly selected seed will have red flowers?
33. The probability that a randomly selected person will be hospitalized within the next month is 0.01. What is the probability that out of three people randomly selected on the street, exactly one will be admitted to the hospital over the next month?
34. A milkmaid serves 4 cows. The probability of getting mastitis within a month for the first cow is 0.1, for the second – 0.2, for the third – 0.2, for the fourth – 0.15. Find the probability that at least one cow will get mastitis within a month.
35. Four hunters agreed to take turns shooting at game. The next hunter fires a shot only if the previous one misses. The probabilities of each hunter hitting the target are the same and equal to 0.8. Find the probability that three shots will be fired.
36. A student is studying chemistry, mathematics and biology. He estimates that the probabilities of getting an A in these courses are 0.5, 0.3, and 0.4, respectively. Assuming that the grades in these courses are independent, find the probability that he will not receive a single “excellent” grade.
37. The student knows 20 out of 25 questions in the program. What is the probability that he knows all three questions of the program proposed to him by the examiner?
38. Two hunters shoot at a wolf, each firing one shot. The probabilities of the first and second hunter hitting the target are 0.7 and 0.8, respectively. What is the probability of hitting the wolf with at least one shot?
39. The probability of hitting the target with three shots at least once for some shooter is 0.875. Find the probability of a hit with one shot.
40. Highly productive cows are selected from the herd. The probability that a randomly selected animal will be highly productive is 0.2. Find the probability that out of three selected cows, only two will be highly productive.
41. In the first cage there are 3 white and 4 gray rabbits, in the second cage there are 7 white and 5 black rabbits. One rabbit was taken at random from each cage. What is the probability that both rabbits are white?
42. The effectiveness of two vaccines was studied in a group of animals. Both vaccines can cause allergies in animals with an equal probability of 0.2. Find the probability that vaccines will not cause allergies.
43. There are three children in the family. Assuming the events of the birth of a boy and a girl to be equally probable, find the probability that all children in the family are of the same sex.
44. The probability of establishing stable snow cover in a given area from October is 0.1. Determine the probability that in the next three years, stable snow cover will be established in this area at least once since October.
45. Determine the probability that a product chosen at random is first-class if it is known that 4% of all products are defective, and 75% of non-defective products meet the requirements of first class.
46. Two shooters, for whom the probabilities of hitting the target are 0.7 and 0.8, respectively, fire one shot each. Determine the probability of at least one hit on the target.
47. The probability of an event occurring in each experiment is the same and equal to 0.2. Experiments are carried out sequentially until the event occurs. Determine the probability that you will have to make a fourth experiment.
48. The probability that the part produced on the first machine will be first-class is 0.7. When manufacturing the same part on a second machine, this probability is 0.8. The first machine produced two parts, the second three. Find the probability that all parts are first-class.
49. A break in the electrical circuit can occur when an element or two elements and fail, which fail independently of each other, respectively, with probabilities of 0.3; 0.2 and 0.2. Determine the probability of an electrical circuit break.
50. The operation of the device stopped due to the failure of one lamp out of 10. Finding this lamp is done by replacing each lamp with a new one in turn. Determine the probability that 7 lamps will have to be checked if the probability of failure of each lamp is 0.1.
51. The probability that the voltage in an electrical circuit will exceed the nominal value is 0.3. At increased voltage, the probability of an accident of the consumer device electric current equal to 0.8. Determine the probability of a device failure due to increased voltage.
52. The probability of hitting the first target for a given shooter is 2/3. If a hit is recorded on the first shot, then the shooter gets the right to shoot at another target. The probability of hitting both targets with two shots is 0.5. Determine the probability of hitting the second target.
53. With the help of six cards, on which one letter is written, the word “carriage” is composed. The cards are shuffled and then taken out one at a time. What is the probability that the word "rocket" is formed in the order in which the letters appear?
54. The subscriber has forgotten the last digit of the phone number and therefore dials it at random. Determine the probability that he will have to call no more than three places.
55. Each of the four incompatible events can occur respectively with probabilities of 0.012; 0.010; 0.006 and 0.002. Determine the probability that at least one of these events will occur as a result of the experiment.
56. What is the probability of drawing a figure of any suit or a card of spades from a deck of 52 cards (the figure is called a jack, queen or king)?
57. The box contains 10 coins of 20 kopecks, 5 coins of 15 kopecks. and 2 coins of 10 kopecks. 6 coins are taken at random. What is the probability that the total will be no more than one ruble?
58. There are balls in two urns: in the first there are 5 white, 11 black and 8 red, and in the second there are 10, 8 and 6, respectively. One ball is drawn at random from both urns. What is the probability that both balls are the same color?
59. The probability for a given athlete to improve his previous result in one attempt is 0.4. Determine the probability that an athlete will improve his result at a competition if two attempts are allowed.
Option 9
1. Each of 6 identical cards has one of the following letters printed on them: o, g, o, r, o, d. The cards are thoroughly mixed. Find the probability that, by placing them in a row, it will be possible to read the word “vegetable garden”.
2. The probability for a given athlete to improve his previous result in 1 attempt is 0.6. Determine the probability that an athlete will improve his result at a competition if he is allowed to make 2 attempts.
3. The first box contains 20 parts, of which 15 are standard; in the second - 30 parts, of which 24 are standard; in the third there are 10 parts, of which 6 are standard. Find the probability that a part taken at random from a box taken at random is standard.
4. Solve problems using Bernoulli’s formula and the Moivre-Laplace theorem: a) when transmitting a message, the probability of distortion of 1 character is 0.24. Determine the probability that a 10-character message contains no more than 3 distortions;
b) 400 trees were planted. The probability that an individual tree will take root is 0.8. Find the probability that the number of surviving trees: 1) is 300; 2) more than 310, but less than 330.
5. Using tabular data, calculate the mathematical expectation, dispersion and standard deviation of the random variable X, and also determine the probability that the random variable will take a value greater than expected.
|
Xi |
||||||
|
P i |
6. Continuous random variable X is specified by the distribution function
Find: a) parameter k; b) mathematical expectation; c) dispersion.
7. A sociological organization conducts a survey of enterprise employees in order to determine their attitude to the structural reorganization carried out by the enterprise management. Assuming that the proportion of people satisfied with structural transformations is described by a normal distribution law with parameters a = 53.1% and σ = 3.9%, find the probability that the proportion of people satisfied with the transformations will be below 50%.
8. A sample was extracted from the general population, which is presented in the form of an interval variation series (see table): a) assuming that the general population has a normal distribution, construct a confidence interval for the mathematical expectation with a confidence probability of γ = 0.95; b) calculate the coefficients of skewness and kurtosis using a simplified method, and make appropriate assumptions about the form of the distribution function of the population; c) using the Pearson criterion, test the hypothesis about the normality of the distribution of the population at a significance level of α = 0.05.
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29-32 |
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|
32-35 |
|
|
35-38 |
|
|
38-41 |
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41-44 |
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44-47 |
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|
47-50 |
9. Given a correlation table of values X and Y: a) calculate the correlation coefficient r xy , draw conclusions about the relationship between X and Y; b) find the linear regression equations of X on Y and Y on X, and also construct their graphs.
|
5.24-5.35 |
5.35-5.46 |
5.46-5.47 |
5.47-5.68 |
5.68-5.79 |
5.79-5.90 |
5.90-6.01 |
6.01-6.12 |
6.12-6.23 |
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21.3-22.0 |
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22.0-22.7 |
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22.7-23.4 |
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23.4-24.1 |
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24.1-24.8 |
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24.8-25.5 |
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25.5-26.2 |
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26.2-26.9 |
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