Probability Notes PDF

1. The number of all possible permutations o o o o 2. How many two-place numbers can be made of the digits 1, 4, 5 and 7 if each digit is included into the image of a number only once? o 24 12 o 16 o 3 o 2 3. The number of all possible allocations o o o o 4. The number of all po...

1. The number of all possible permutations o o o o 2. How many two-place numbers can be made of the digits 1, 4, 5 and 7 if each digit is included into the image of a number only once? o 24 12 o 16 o 3 o 2 3. The number of all possible allocations o o o o 4. The number of all possible combinations o o o o 5. How many ways are there to choose 2 details from a box containing 9 details? o 12 o 4 o 22 o 11 36 6. The numbers of allocations, permutations and combinations are connected by the equality  o o o o 7. If some object A can be chosen from the set of objects by m ways, and another object B can be chosen by n ways, then we can choose either A or B by … ways. m+n o m-n o n-m o n! o 8. Events are equally possible if … o none of them will necessarily happen as a result of a trial there is reason to consider that none of them is more possible (probable) than other o there is reason to consider that one of them is more possible (probable) than other o at least one of them will necessarily happen as a result of a trial o one of them will necessarily happen as a result of a trial 9. The probability of the event A is determined by the formula o , where is the space of elementary outcomes o , where is the space of elementary outcomes , where is the space of elementary outcomes o where is the space of elementary outcomes o , where is the space of elementary outcomes 10. The probability of a reliable event is equal to … 1 o 0 o ½ o 1/3 o 1/5 11. The probability of an impossible event is equal to … 0 o 1 o ½ o 1/3 o 1/5 12. The probability of a random event is … the positive number between 0 and 1 o the positive number between 0 and ½ o the positive number between 0 and 10 o the positive number between 0 and 1/3 o the positive number between 0 and 1/5 13. The relative frequency of the event A is defined by the formula: , where m is the number of appearances of the event, n is the total number of trials. o , where n is the number of appearances of the event, m+1 is the total number of trials. o , where m+1 is the number of appearances of the event, n is the total number of trials. o , where m is the number of appearances of the event, n is the total number of trials. o , where m is the number of appearances of the event, n is the total number of trials. 14. There are 100 identical details (and 20 of them are painted) in a box. Find the probability that the first randomly taken detail will be painted. o 1/20 1/5 o ½ o 1/10 o 1/9 15. A die is tossed. Find the probability that an even number of aces will appear. ½ o 1 o 0 o 1/5 o 1/9 16. Participants of a toss-up pull a ticket with numbers from 1 up to 30 from a box. Find the probability that the number of the first randomly taken ticket contains the digit 2. o 1/30 o 1/3 o ½ 2/5 o 1/5 17. In a batch of 8 details the quality department has found out 3 non-standard details. What is the relative frequency of appearance of non-standard details equal to? o ½ o 1 o 3/11 3/8 o 3/5 18. At shooting by a rifle the relative frequency of hit in a target has appeared equal to 0,4. Find the number of hits if 20 shots were made. 8 o 3 o 20 o 1 o 6 19. Two dice are tossed. Find the probability that different number of aces will appear on dices o 1/6 5/6 o ½ o 1/3 o 1 20. Two dice are tossed. Find the probability that the sum of aces will exceed 10. 1/12 o 5/12 o 5/18 o 1/18 o 0 21. An urn contains 15 balls: 4 white, 6 black and 5 red. Find the probability that a randomly taken ball will be red or white. o 3/7 o 4/15 3/5 o 6/15 o 1 22. 12 seeds have germinated of 60 planted seeds. Find the relative frequency of germination of seeds. 1/5 o 4/5 o 1/60 o 1/12 o 1 23. A point C is randomly appeared in a segment AB of the length 5. Determine the probability that the distance between C and B doesn’t exceed 1. o 2/5 1/5 o 4/5 o 0 o ½ 24. A coin is tossed twice. Find the probability that the coin lands on tails in both times. 1/4 o 3/4 o 1/3 o ½ o 0 25. There are 200 details in a box. It is known that 150 of them are details of the first kind, 10 – the second kind, and the rest – the third kind. How many ways of extracting a detail of the first or the third kind from the box are there? 190 o 10 o 40 o 200 o 150 26. If an object A can be chosen from the set of objects by m ways and after every such choice an object B can be chosen by n ways then the pair of the objects (A, B) in this order can be chosen by... ways. mn o m+n o n-m o n! o N 27. There are 12 students in a group. It is necessary to choose a leader, its deputy and head of professional committee. How many ways of choosing them are there? 1320 o 480 o 12 o 144 o 1630 28. 5 of 20 students have sport categories. What is the probability that 3 randomly chosen students have sport categories? 1/114 o 3/5 o 1/10 o 1/57 o 1/1218 29. A box contains 5 red, 6 green and 4 blue pencils. 3 pencils are randomly extracted from the box. Find the probability that all the extracted pencils are different color. o 4/19 o 15/91 24/91 o 3/19 o 3/91 30. It has been sold 12 of 15 refrigerators of three marks available in quantities of 5, 7 and 3 units in a shop. Assuming that the probability to be sold for a refrigerator of each mark is the same, find the probability that refrigerators of one mark have been unsold. 0,101 o 0,016 o 0,984 o 0,53 o 0,8 31. A shooter has made three shots in a target. Let Ai be the event «hit by the shooter at the i-th shot» (i = 1, 2, 3). Express by A1, A2, A3 and their negations the following event A – «only two hit». o o o o 32. The probability of appearance of any of two incompatible events is equal to: o o o o 33. There are 20 balls in an urn: 3 red, 2 blue and 15 white. Find the probability of appearance of a color (red or blue) ball. 1/4 o 1 o 3/4 o 1/20 o 1/15 34. A shooter shoots in a target subdivided into three areas. The probability of hit in the first area is 0,5 and in the second – 0,3. Find the probability that the shooter will hit at one shot either in the second area or in the third area. 0,5 o 0,9 o 1 o 0,7 o 0,8 35. The sum of the probabilities of events A1, A2,…, An which form a complete group is equal to … 1 o 0 o ½ o 1/5 o 1/3 36. A consulting point of an institute receives packages with control works from the cities A, B and С. The probability of receiving a package from the city A is equal 0,2; from the city B – 0,2. Find the probability that next package will be received from the city С. o 0,5 o 0,9 0,6 o 0,8 o 1 37. Two uniquely possible events forming a complete group are … Opposite o Same o Identically distributed o Sample o Density function 38. The sum of the probabilities of opposite events is equal to … 1 o 0 o ½ o 1/3 o 1/5 39. The probability that a day will be rainy is p = 0,75. Find the probability that a day will be clear. 0,25 o 0,3 o 0,15 o 0,75 o 1 40. The conditional probability of an event A with the condition that an event B has already happened is equal to: o o o o 41. There are 4 conic and 8 elliptic cylinders at a collector. The collector has taken one cylinder, and then he has taken the second cylinder. Find the conditional probability that the second taken cylinder is elliptic given that the first was conic. 8/11 o 1/4 o 8/33 o 1/2 o 2/3 42. There are 4 white, 5 black and 6 blue balls in an urn. Each trial consists in extracting at random one ball without replacement. Find the probability that a white ball will appear at the first trial (the event A), a black ball will appear at the second trial (the event B), and a blue ball will appear at the third trial (the event C). o 1/22 4/91 o 1/8 o 8/225 o ½ 43. The events A, B, C and D form a complete group. The probabilities of the events are those: P(A) = 0,1; P(B) = 0,49; P(C) = 0,3. What is the probability of the event D equal to?  0,11 o 0,5 o 0,2 o 0,4 o 0,1 44. For independent events theorem of multiplication has the following form: o o o o 45. Find the probability of joint hit in a target by two guns if the probability of hit in the target by the first gun (the event A) is equal to 0,3; and by the second gun (the event B) – 0,5. o 0,35 0,15 o 0,2 o 0,8 o 0,5 46. There are 3 boxes containing 10 details each. There are 5 standard details in the first box, 6 – in the second and 3 – in the third box. One takes at random on one detail from each box. Find the probability that all three taken details will be standard. 0,09 o 0,5 o 0,14 o 0,49 o 0,27 47. The probabilities of hit in a target at shooting by three guns are the following: p1 = 0,6; p2 = 0,7; p3 = 0,5. Find the probability of at least two hits at one shot by all three guns. o 0,874 0,65 o 0,94 o 0,09 o 0,76 48. There are 3 flat-printing machines at typography. For each machine the probability that it works at the present time is equal to 0,6. Find the probability that at least one machine works at the present time. o 0,9714 o 0,256 o 0,36 o 0,784 0,936 49. What is the probability that at tossing two dice 3 aces will appear at least on one of the dice? 0,306 o 0,278 o 0,421 o 0,529 o 0,386 50. Three shots are made in a target. The probability of hit at each shot is equal to 0,6. Find the probability that only two hits will be in result of these shots. 0,432 o 0,204 o 0,512 o 0,288 o 0,592 51. Three students pass an exam. The probability that the exam will be passed on "excellent" by the first student is equal to 0,5; by the second – 0,2; and by the third – 0,8. What is the probability that the exam will be passed on "excellent" by only one student? 0,42 o 0,48 o 0,92 o 0,28 o 0,99 52. Three students pass an exam. The probability that the exam will be passed on "excellent" by the first student is equal to 0,5; by the second – 0,3; and by the third – 0,7. What is the probability that the exam will be passed on "excellent" by exactly two students? o 0,464 0,395 o 0,12 o 0,192 o 0,48 53. Three students pass an exam. The probability that the exam will be passed on "excellent" by the first student is equal to 0,3; by the second – 0,7; and by the third – 0,8. What is the probability that the exam will be passed on "excellent" by at least one student? 0,958 o 0,93 o 0,465 o 0,15 o 0,848 54. Three students pass an exam. The probability that the exam will be passed on "excellent" by the first student is equal to 0,3; by the second – 0,7; and by the third – 0,8. What is the probability that the exam will be passed on "excellent" by neither of the students? 0,042 o 0,95 o 0,46 o 0,07 o 0,84 55. Three buyers went in a shop. The probability that each buyer makes purchases is equal to 0,8. Find the probability that two of them will make purchases. 0,384 o 0,7 o 0,189 o 0,96 o 0,904 56. Four buyers went in a shop. The probability that each buyer makes purchases is equal to 0,5. Find the probability that three of them will make purchases. 0,25 o 0,096 o 0,95 o 0,125 o 0,712 57. Four buyers went in a shop. The probability that each buyer makes purchases is equal to 0,8. Find the probability that only one of them will make purchases. 0,0256 o 0,568 o 0,0441 o 0,064 o 0,144 58. There are 5 details made by the factory № 1 and 15 details of the factory № 2 at a collector. Two details are randomly taken. Find the probability that at least one of them has been made by the factory № 1. 17/38 o 5/19 o 16/19 o 1/5 o 1/16 59. There are 5 details made by the factory № 1 and 15 details of the factory № 2 at a collector. Two details are randomly taken. Find the probability that at least one of them has been made by the factory № 2. 18/19 o 5/19 o 16/19 o 1/5 o 1/16 60. 10 of 20 savings banks are located behind a city boundary. 4 savings banks are randomly selected for an inspection. What is the probability that among the selected banks appears inside the city 2 savings banks? 0.418 o 0.07 o 0.5 o 0.6 o 0.23 61. The probabilities that three men hit a target are respectively 1/3, 1/4 and 1/2. Each man shoots once at the target. What is the probability that exactly one of them hits the target? o 19/40 o 21/24 11/24 o 3/4 o 17/72 62. A problem in mathematics is given to three students whose chances of solving it are 2/3, 3/4, 2/5. What is the probability that the problem will not be solved? o 19/20 o 17/20 o 15/20 1/20 o 1 63. A problem in mathematics is given to three students whose chances of solving it are 1/3, 3/4, 3/5. What is the probability that the problem will be solved? o 9/15 o 13/15 o 3/4 14/15 o 1 64. There are two sets of details. The probability that a detail of the first set is standard is equal to 0,7; and of the second set – 0,4. Find the probability that a randomly taken detail (from a randomly taken set) is standard. o 0.8 o 0.5 o 0.45 0.55 o 0.2 65. There are two sets of details. The probability that a detail of the first set is standard is equal to 0,7; and of the second set – 0,4. Find the probability that a randomly taken detail (from a randomly taken set) is not standard. o 0.8 o 0.5 0.45 o 0.55 o 0.2 66. An urn contains 10 balls: 3 red and 7 blue. A second urn contains 6 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.54. Calculate the number of blue balls in the second urn. 9 o 2 o 4 o 5 o 6 67. The probability that a boy will not pass M.B.A. examination is 1/5 and that a girl will not pass is 3/5. Calculate the probability that at least one of them passes the examination. o 11/25 o 13/25 o 1/2 22/25 o 16/25 68. The probability that a boy will not pass M.B.A. examination is 1/5 and that a girl will not pass is 3/5. Calculate the probability that exactly one of them passes the examination. o 11/25 o 22/25 o 1/2 14/25 o 16/25 69. A bag contains 6 red discs and 4 blue discs. If 3 discs are drawn from the bag without replacement, find the conditional probability that all three will be blue given that one of them is blue. 1/33 o 5/33 o 11/16 o 4/165 o 1/3 70. A bag contains 4 white, 6 red and 10 black balls. Four balls are drawn one by one with replacement, what is the probability that at least one is white? 4 1 o 1−   4 4 1 1 −   5 4 1 o   5 o 0.7182 4 1 o   4 71. Which of the following expressions indicates the occurrence of exactly one of the events A, B, C? o A+ B+C o A⋅ B ⋅C A ⋅ B c ⋅ C c + Ac ⋅ B ⋅ C c + Ac ⋅ B c ⋅ C o ( A + B + C)c o AB + AC + BC 72. A fair die is rolled three times. A random variable X denotes the number of occurrences of 6’s. What is the probability that X will have the value which is not equal to 3. o 91/216 o 125/216 o 25/216 o 1/216 215/216 73. If a fair die is tossed twice, the probability that the first toss will be a number less than 3 and the second toss will be greater than 5 is o 1/3 o 5/18 1/18 o 3/18 o 0 74. A class consists of 460 female and 540 male students. The students are divided according to their marks. If one person is selected randomly, the probability that it did not pass given that it is female is: o 0.06 0.13 o 0.15 o 0.101 o none of the shown answers 75. How many different two-member teams can be formed from six students? 15 o 12 o 21 o 72 o 3 76. How many different 3-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? o 6 o 36 120 o 46.656 o 72 77. If P(E) is the probability that an event will occur, which of the following must be false? o P(E)=1 o P(E)=1/2 o P(E)=1/3 P(E)= - 1/3 o P(E)=0 78. A die is rolled. What is the probability that the number rolled is greater than 3 and even? o 1/2  1/3 o 2/3 o 5/6 o 0 79. How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? o 6 o 36 720 o 46 o 1440 80. Evaluate 0!+1!+4! o 5 o 13 26 o 25 o 24 81. Evaluate 6!-5! o 5 o 6 600 o 25 o 24 82. Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license plates can be issued with two letters followed by two numbers? o 2500 o 6760 o 25000 67600 o 2500 83. A fair coin is thrown in the air five times. If the coin lands with the head up on the first four tosses, what is the probability that the coin will land with the head up on the fifth toss? o 0 o 1/16 o 1/8 1/2 o 1/4 84. A movie theatre sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? o 1 o 3 9 o 27 o 62 85. Two events each have probability 0.3 of occurring and are independent. The probability that neither occur is 0.49 o 0.51 o 0.3 o 0.6 o none of the given answers 86. A class consists of 490 female and 510 male students. The students are divided according to their marks Passed and Did not pass. If one person is selected randomly, what is the probability that it did not pass given that it is male. o 0.17 o 0.21 o 0.42 o 0.08 0.196 87. A student can solve 6 from a list of 10 problems. For an exam 8 questions are selected at random from the list. What is the probability that the student will solve exactly five problems? o 0.98 o 0.02 o 0.28 0.53 o None of the shown answers 88. Suppose that 10% of people are left handed. If 6 people are selected at random, what is the probability that exactly 2 of them are left handed? o 0.031 o 0.053 0.098 o 0.0100 o 0.29 89. Suppose a computer chip manufacturer rejects 15% of the chips produced because they fail presale testing. If you test 4 chips, what is the probability that not all of the chips fail? 0.9995 o 0,00005 o 0.15 o 0.6 o 0.5220 90. Which of the following is the appropriate definition for the union of two events A and B? o The set of all possible outcomes. o The set of all basic outcomes contained within both A and B. The set of all basic outcomes in either A or B, or both. o None of the given answers o The set of all basic outcomes that are not in A and B. 91. Johnson taught a music class for 20 students under the age of ten. He randomly chose one of them. What was the probability that the student was under eleven? 1 o 0.5 o 1/25 o 0 o 0.25 92. The compact disk Jane bought had 12 songs. The first five were rock music. Tracks number 6 through 12 were ballads. She selected the random function in her CD Player. What is the probability of first listening to a ballad? o 1/3 o 2/3 o 5/12 o 1/6 7/12 93. Two fair dice, one red and one blue, each have numbers 1-6. If a roll of the two dice totals 6, what is the probability that the red die is showing a 3? o 1/6 1/5 o 1/3 o 5/6 o 1/18 94. A regular deck of 52 cards contains 4 different suits (Spades, Hearts, Diamonds, and Clubs) that each have 13 cards. If you randomly choose two cards from the deck, what is the probability that both cards will all be Spades? 1/17 o 2/17 o 1/4 o 4/17 o 33/68 95. A standard deck of 52 cards contains 4 different suits (Spades, Hearts, Diamonds, and Clubs) that each have 13 cards. What is the probability of drawing a Diamond from a standard deck of 52 cards? o 1/52 o 13/39 o 1/13 1/4 o 1/2 96. One card is randomly selected from a shuffled deck of 52 cards and then a die is rolled. Find the probability of obtaining an Ace and rolling an odd number. o 1/104 o 7/13 o 1/39 1/26 o 1/36 97. In the first step, Joe draws a hand of 5 cards from a deck of 52 cards. What is the probability that Joe has exactly one ace? 0.2995 o 0.699 o 0.23336 o 1/4 o 0.4999