Probability theory manual. Ministry of Education of the Russian Federation Kazan State Technical University named after V.I. A. N. Tupoleva Theory of Probability (Tutorial). Tasks for self-solving Probability for the athlete will improve
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Lesson 4. The theorem of addition of probabilities.
14.1. Brief theoretical part
The probability of the sum of two events is determined by the formulaP( A+AT) = 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 in the course of 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 at the cinema
c) The onset of spring according to the calendar and snowfall
d) The appearance on the dropped face of each of the two dice of three points and the equality of the sum of points on the dropped faces of both dice to an odd number
e) Showing a football match on one television channel and a news release on another
The addition theorem for the probabilities of incompatible events is formulated as follows:
b) The probability of occurrence of one of two incompatible events is equal to the sum of the probabilities of these events
c) The probability of occurrence of one of two incompatible events is equal to the difference between the probabilities of occurrence of these events
The addition theorem for joint event probabilities is formulated as follows:
b) The probability of the occurrence of at least one of the 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 the two joint events is equal to the sum of the probabilities of these events and the probability of their joint occurrence
The probability addition theorem is generalized to the sum of any number of events, and the probability of the sum of events in general is calculated by the formula:
If the events are incompatible, then the probability of the sum of these events is equal to:
b)
in)
34.3. Solution of typical tasks
Example 4.1. Determine the probability that a batch of one hundred products, including five defective ones, will be accepted when testing at random a selected half of the entire batch, if the acceptance conditions allow no more than one out of fifty defective products.Decision.
With, consisting in the fact that a batch of one hundred products, among which five are defective, will be accepted when testing at random a selected half of the entire batch.
Denote by BUT an event consisting in the fact that no defective products were received during the test, and after AT- an event consisting in the fact that only one defective item is received.
Since С=А+В, then the desired probability P(C) = Р( BUT+B).
Events BUT and AT incompatible. Therefore P(C) = P( BUT)+ P( B).
Of the 100 products, 50 can be selected in ways. Of the 95 non-defective products, 50 can be selected in ways.
Therefore R( A)=.
Similarly R( B)= .
P(C) = P( BUT)+ P( B)=+==0,181.
Example 4.2.
Electrical circuit between points M and N compiled according to the scheme shown in Fig. 5.
Failure over time T different elements of the chain - independent events with 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 breaking in a specified time interval.
Decision.
Let us consider the event With, consisting in the fact that for the specified period of time there will be a chain break.
Denote by A j (j= 1,2) an event consisting in the failure of an element To j, through BUT- failure of at least one element To j, and through AT- failure of all three elements BUT i (i=1, 2, 3).
Then the desired probability
R( With) = P( A + AT) = P( A) + P( AT) - R( A)R( B).
R( A) = P( A 1 ) + P( A 2 ) - R( A 1 )R( A 2 ) = 0,8,
R( AT) = P( L 1 )R( L 2 ) R( L 3 ) = 0,252,
then.
Example 4.3.
The urn contains n white, m blacks and l red balls, which are drawn at random one at a time:
a) no return
b) with return after each extraction.
Determine in both cases the probabilities that the white ball will be drawn before the black one.
Decision.
Let be R 1 is the probability that the white ball will be drawn before the black one, and R 11 is 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 reds, etc. Thus, one can write in the case when the balls are not returned,
and when the balls return
To get probabilities R 11
in the previous formulas, you need to replace n on the m, a m on the n. It follows that in both cases R 1
:R 11
= n:m. Since, moreover, R 1
+R 11
= 1, then the desired probability when drawing balls without replacement is also equal.
Example 4.4.
Someone wrote n letters, sealed them in envelopes, and then randomly wrote a different address on each of them. Determine the probability that at least one of the envelopes has the correct address.
Decision.
Let the event A k is that on k-th envelope contains the correct address ( k=l, 2,..., n).
The desired probability.
Events A k joint; for any different k, j, i, ... 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 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 hitting the target.
(Answer:
p = 0.55)
4.3. Two identical coins of radius 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 piece of any suit or a card of spades from a deck of 52 cards (a piece 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 drawn at random. What is the probability that they will total at most 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 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 conducted under the following conditions: as a result of the first move, which always makes BUT, he can win with a probability of 0.3; if the first move A does not win, then the move is made AT and can win with a probability of 0.5; if as a result of this move AT doesn't win then A makes a second move, which can lead to his winning with a probability of 0.4. Determine the winning probabilities for BUT and for AT.
(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 performance in a competition if two attempts are allowed.
(Answer:
p(A) =)
4.9. From an urn containing n balls numbered from 1 to n, two balls are drawn in succession, with the first ball being returned if its number is not equal to one. Determine the probability that the ball with number 2 will be drawn on the second draw.
(Answer:
p =)
4.10. Player BUT alternately playing with the players AT 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 probability of winning AT and With.
(Answer:
)
4.11. Two people take turns tossing a coin. The one with the coat of arms first wins. Determine the probability of winning for each of the players.
(Answer:
)
4.12. The probability of getting a point without losing a serve, when playing two equivalent volleyball teams, is half. Determine the probability of getting one point for the serving team.
(Answer:
p =)
4.13. Two shooters alternately shoot at the target until the first hit. The hit probability for the first shooter is 0.2, and for the second shooter it is 0.3. Find the probability that the first shooter fires more shots than the second.
(Answer:
p = 0.455)
4.14. Two play to win, and for this it is necessary for the first to win t parties, and the second P parties. The probability of winning each game by the first player is equal to R, and the second q=1-R. Determine the probability of winning the whole game by the first player.
(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 fire independently at the target. 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 arrows 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 balls are white?
6. Fire three shots at one target. The probability of hitting each shot is 0.5. Find the probability that only one hit will occur as a result of these shots.
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 a part made on the first machine will be first-class is 0.7. If the same part is made on the second machine, this probability is 0.8. On the first machine two parts are made, on 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 . The search for this lamp is carried out by replacing each lamp with a new one in turn. Determine the probability that you have to check 2 lamps, if the probability of failure of each lamp is p = 0.2 .
10. On the site AB There are 12 obstacles for a racing motorcyclist, the probability of stopping at each of them is 0.1. The probability that from the point AT to the final destination With the motorcyclist will pass without stopping, is equal to 0.7. Determine the probability that the area AC there will be no stopping.
11. There are 4 traffic lights on the way of the car. The probability of stopping at the first two is 0.3, and the next two are 0.4. What is the probability of passing traffic lights without stopping?
12. There are 3 traffic lights on the way of the car. The probability of stopping at the first two is 0.4, and at the third 0.5. What is the probability of passing traffic lights with one stop?
13. Two network servers on the Internet are at 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 at the first shot, then the shooter gets the right to the second. If at the second he hits again, then he shoots a third time. What is the probability of hitting with three shots?
15. Game between BUT and AT is played under the following conditions: as a result of the first move, which always makes BUT, he can win with a probability of 0.3; if the first move BUT does not win, then the move is made AT and can win with a probability of 0.5; if as a result of this move AT doesn't win then BUT makes a second move, which can lead to his winning with a probability of 0.4. Determine the winning probabilities for BUT and for AT.
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 performance in a competition if two attempts are allowed.
17. Player BUT alternately plays two games with the players AT and WITH. Probabilities of winning the first game for AT and With are equal to 0.1 and 0.2, respectively; the probability of winning in the second game for AT is 0.3, for With equals 0.4. Determine the probability that: a) B wins first; b) to win first WITH.
18. From an urn containing P balls numbered from 1 to n, two balls are drawn in succession, the first being returned if its number is not equal to one. Determine the probability that the ball with number 2 will be drawn on the second draw.
19. Player BUT plays alternately with players B and C, with a probability of winning in each set 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. which the coat of arms will appear first. Determine the probability of winning for each of the players.
21. There are 8 white and 6 black balls in an urn. Two players draw one ball in succession, each time returning the drawn ball. The game continues until one of them gets a white ball. Determine the probability that the player who starts the game draws a white ball first.
22. A courier was sent for documents in 4 archives. The probability of the presence of the necessary documents in the I-th archive is 0.9; in II - 0.95; in III-em - 0.8; in IV - ohm - 0.6. Find the probability P of the absence of a document in only one archive.
23. Find the probability that two of the three independently operating elements of the computing device will fail if the probability of failure of the first, second and third elements, respectively, is 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 exchange of mineral salts. Three animals are taken consecutively without return. What is the probability that they are healthy?
26. The pond contains 12 crucians, 18 breams and 10 carps. Caught three fish. Find the probability that two carp and crucian were caught in succession.
27. There are 12 cows in the herd, 4 of them are of the Simmental breed, the rest are of the Hallstein-Friest breed. Three animals were selected for selection work. Find the probability that all three of them are Simmental breeds.
28. At the hippodrome there are 10 bay horses, 3 gray ones and 7 white ones. 2 horses were randomly selected for the race. What is the probability that there is no white horse among them?
29. There are 9 dogs in the kennel, 3 of them are collies, 2 are boxers, the rest are dogs. Three dogs are randomly selected. What is the probability that there is at least one boxer among them?
30. The average offspring of animals is 4. The appearance of female and male individuals is equally probable. Find the probability that there are two males in the offspring.
31. The package contains seeds, the germination of which is 0.85. The probability that the plant will flower is 0.9. What is the probability that a plant grown from a randomly selected seed will flower?
32. The package contains bean seeds, the germination rate of which is 0.9. The probability that 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 during the 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 shoot at the game in turn. The next hunter fires a shot only if the previous one misses. The probabilities of hitting the target by each of the hunters are the same and equal to 0.8. Find the probability that three shots will be fired.
36. A student studies chemistry, mathematics and biology. He estimates that the probabilities of getting "excellent" 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 any "excellent" grades.
37. The student knows 20 of the 25 questions of the program. What is the probability that he knows all three questions of the program given to him by the examiner?
38. Two hunters shoot at a wolf, and each makes one shot. The probabilities of hitting the target by the first and second hunters are 0.7 and 0.8, respectively. What is the probability of hitting a 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 hitting 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 only two out of three selected cows 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 efficacy of two vaccines was studied in a group of animals. Both vaccines can cause allergy in animals with equal probabilities of 0.2. Find the probability that the vaccines will not cause an allergy.
43. There are three children in the family. Assuming the events consisting in the birth of a boy and a girl are equally likely, find the probability that all children in the family are of the same sex.
44. The probability of establishing a stable snow cover in a given area since October is 0.1. Determine the probability that in the next three years in this area a stable snow cover will be established 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 the first grade.
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 equals 0.2. Experiments are performed sequentially until the event occurs. Determine the probability that a fourth experiment will have to be performed.
48. The probability that the part made on the first machine will be first-class is 0.7. In the manufacture of the same part on the second machine, this probability is 0.8. On the first machine two parts are made, on 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 fail and which fail independently of each other, respectively, with probabilities of 0.3; 0.2 and 0.2. Determine the probability of breaking the electrical circuit.
50. The operation of the device stopped due to the failure of one lamp out of 10. The search for this lamp is carried out 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 the electrical circuit will exceed the nominal value is 0.3. At increased voltage, the probability of an accident of the device - the consumer of electric current is 0.8. Determine the probability of device failure due to voltage increase.
52. The probability of hitting the first target for a given shooter is 2/3. If a hit is recorded during 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 randomly drawn one at a time. What is the probability that the word "rocket" is formed in the order of the letters?
54. The subscriber forgot the last digit of the phone number and therefore dials it at random. Determine the probability that he will have to call at most 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 extracting from a deck of 52 cards a piece of any suit or a card of spades (a piece is called a jack, queen or king)?
57. There are 10 coins of 20 kopecks in a box, 5 coins of 15 kopecks each. and 2 coins of 10 kopecks. 6 coins are taken at random. What is the probability that they will total at most one ruble?
58. There are balls in two urns: in the first 5 white, 11 black and 8 red, and in the second 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 performance in a competition if two attempts are allowed.
4.1. Each of the four incompatible events can occur with probabilities 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 hitting the target.
(Answer: p = 0.55)
4.3. Two identical coins of radius r are placed inside a circle of radius R into which a dot 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 piece of any suit or a card of spades from a deck of 52 cards (a piece 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 drawn at random. What is the probability that they will total at most 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 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 a probability of 0.3; if A does not win on the first move, then B makes the move and can win with a probability of 0.5; if as a result of this move B does not win, then A makes a second move, which can lead to his winning with a probability of 0.4. Determine the winning probabilities 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 performance in a competition if two attempts are allowed.
(Answer: p(A) = )
4.9. From an urn containing n balls numbered from 1 to n, two balls are drawn in succession, with the first ball being returned if its number is not equal to one. Determine the probability that the ball with number 2 will be drawn on the second draw.
(Answer: p = )
4.10. Player A alternates with players B and C, with a probability of winning each set of 0.25, and stops the game 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 with the coat of arms first wins. Determine the probability of winning for each of the players.
(Answer: 
)
4.12. The probability of getting a point without losing a serve, when playing two equivalent volleyball teams, is half. Determine the probability of getting one point for the serving team.
(Answer: p = )
4.13. Two shooters alternately shoot at the target until the first hit. The hit probability for the first shooter is 0.2, and for the second shooter it is 0.3. Find the probability that the first shooter fires more shots than the second.
(Answer: p = 0.455)
4.14. Two play until victory, and for this it is necessary for the first to win m games, and for the second n games. The probability of winning each game by the first player is p, and the second is q=1-p. Determine the probability of winning the whole game by the first player.
Option 9
1. One of the following letters is printed on each of 6 identical cards: o, g, o, p, 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 "garden".
2. The probability for a given athlete to improve his previous result from 1 attempt is 0.6. Determine the probability that in a competition an athlete will improve his result if 2 attempts are allowed.
3. The first box contains 20 parts, 15 of them are standard; in the second - 30 parts, of which 24 are standard; in the third - 10 parts, of which 6 are standard. Find the probability that a randomly selected part from a randomly taken box is a standard one.
4. Solve problems using the Bernoulli formula and the Moivre-Laplace theorem: a) when transmitting a message, the probability of distortion of 1 sign is 0.24. Determine the probability that a message of 10 characters contains no more than 3 distortions;
b) 400 trees were planted. The probability that an individual tree will survive is 0.8. Find the probability that the number of surviving trees: 1) equals 300; 2) more than 310 but less than 330.
5. Using tabular data, calculate the mathematical expectation, variance and standard deviation of the random variable X, and also determine the probability that the random variable will take on a value greater than expected.
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Х i |
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P i |
6. A continuous random variable X is given by the distribution function
Find: a) parameter k ; b) mathematical expectation; c) dispersion.
7. The sociological organization conducts a survey of the employees of the enterprise in order to clarify their attitude towards the structural reorganization carried out by the management of the enterprise. 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 γ = 0.95; b) calculate the coefficients of skewness and kurtosis using a simplified method and make appropriate assumptions about the form of the population distribution function; c) using the Pearson test, test the hypothesis of the normality of the distribution of the general population at a significance level of α = 0.05.
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29-32 |
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32-35 |
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35-38 |
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38-41 |
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41-44 |
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44-47 |
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47-50 |
9. A correlation table of X and Y values is given: a) calculate the correlation coefficient r xy , draw conclusions about the relationship between X and Y; b) find the equations of linear regression X on Y and Y on X, and plot their graphs.
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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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