Probability Theory

Question 1

Bob has three bookshelves in his office and 15 books (5 are math books, 10 are novels). If each shelf holds exactly five books and books are placed randomly on the shelves (all orderings are equally likely), how many ways can 15 books be arranged on a bookshelves?

Accepted Answer

15!/10!5!

Answer

14!/10!5!

Answer

5!10!/15!

Answer

15!10!/5!

Question 2

A class in probability theory consists of 2 men and 8 women. They passed exam, took their score. Assume that no two students took the same score. How many different scores are possible?

Accepted Answer

10!

Answer

6!4!

Answer

10!/2!

Answer

10!/6!4!

Answer

10!/(2!)^2

Question 3

A class in probability theory consists of 6 men and 4 women. They passed exam, took their score. Assume that no two students took the same score. How many different scores are possible?

Accepted Answer

10!

Answer

6!4!

Answer

10!/2!

Answer

10!/6!4!

Answer

10!/(2!)^2

Question 4

Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are math books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible?

Accepted Answer

6912

Answer

288

Answer

1260

Answer

525

Answer

3456

Question 5

How many different signals, each consisting of 9 flags hung in a line, can be made from a set of 4 white flags, 3 red flags, and 2 blue flags if all flags of the same color are identical?

Accepted Answer

1260

Answer

288

Answer

6912

Answer

525

Answer

3456

Question 6

From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed, if 2 of the men are refuse to serve on the committee together?

Accepted Answer

300

Answer

350

Answer

4200

Answer

500

Answer

220

Question 7

A student is to answer to 10 questions in an examination. How many choices if she answered at 7 questions?

Accepted Answer

120

Answer

176

Answer

45

Answer

10

Answer

220

Question 8

A student is to answer to 10 questions in an examination. How many choices if she answered at least 7 questions?

Accepted Answer

176

Answer

120

Answer

45

Answer

10

Answer

220

Question 9

An urn contains 30 balls, of which 10 are red and the other 20 blue. Suppose you take out 8 balls from this urn, without replacement. In how many ways among chosen 8 balls in this sample exactly 3 are red and 5 are blue?

Accepted Answer

1860480

Answer

5852925

Answer

3720960

Answer

2480640

Answer

4961280

Question 10

A bag contains six Scrabble tiles with the letters A-D-M-N-O-R. You reach into the bag and take out tiles one at a time. After you pick a tile from the bag, write down that letter and then return the tile to the bag. How many possible words can be formed?

Accepted Answer

46656

Answer

720

Answer

6

Answer

120

Answer

10240

Question 11

A police department in a small city consists of 10 officers. If the department policy is to have 5 of the officers patrolling the streets, 2 of the officers working full time at the station, and 3 of the officers on reserve at the station, how many different divisions of the 10 officers into the 3 groups are possible?

Accepted Answer

2520

Answer

2025

Answer

2250

Answer

2502

Answer

2052

Question 12

Joel has an MP3 player called the Jumble. The Jumble randomly selects a song for the user to listen to. Joel's Jumble has 2 classical songs, 13 rock songs and 5 rap songs on it. What is the probability that the selected song is rock song or rap song?

Accepted Answer

0.9

Answer

0.1

Answer

13/20

Answer

18

Answer

0.18

Question 13

Joel has an MP3 player called the Jumble. The Jumble randomly selects a song for the user to listen to. Joel's Jumble has 2 classical songs, 13 rock songs and 5 rap songs on it. What is the probability that the selected song is classical song or rap song?

Accepted Answer

0.35

Answer

0.9

Answer

13/20

Answer

7

Answer

0.7

Question 14

A chess tournament has 10 competitors, of which 4 are Russian, 3 are from the USA, 2 are from Great Britain, and 1 is from Brazil. If the tournament result lists just the nationalities of the players in the order in which they placed, how many outcomes are possible?

Accepted Answer

12600

Answer

12400

Answer

12500

Answer

12700

Answer

16200

Question 15

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 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 0.44. Calculate the number of blue balls in the second urn.

Accepted Answer

4

Answer

64

Answer

16

Answer

32

Answer

8

Question 16

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the different color is 0.44. Calculate the number of blue balls in the second urn.

Accepted Answer

64

Answer

4

Answer

32

Answer

16

Answer

8

Question 17

License plates in Minnesota are issued with three letters from A to Z followed by three digits from 0 to 9. If each license plate is equally likely, what is the probability that a random license plate starts with G-Z-N?

Accepted Answer

1/26^3

Answer

26^3

Answer

1/26!3!

Answer

26!3!

Answer

1/3^26

Question 18

A business man has 4 dress shirts and 7 ties. How many different shirt/tie outfits can he create?

Accepted Answer

28

Answer

4

Answer

7

Answer

11

Answer

8

Question 19

Mark is taking four final exams next week. His studying was erratic and all scores A, B, C, D, and F are equally likely for each exam. What is the probability that Mark will get at least one A?

Accepted Answer

0.5904

Answer

0.3264

Answer

0.8712

Answer

0.6124

Answer

0.9122

Question 20

Mark is taking four final exams next week. His studying was erratic and all scores A, B, C, D, and F are equally likely for each exam. What is the probability that Mark will get at least one B?

Accepted Answer

0.5904

Answer

0.3264

Answer

0.8712

Answer

0.6124

Answer

0.9122

Question 21

The Survey of women who are age 18 or older indicated that 86% of married women have one or more children, 31% of never married women have one or more children, and 86% of women who are divorced or widowed have one or more children. The survey also indicated that 48% of women were currently married, 24% had never been married, and 28% were divorced or widowed. If a randomly selected woman has one or more children, what is the probability that she is married?

Accepted Answer

0.567

Answer

0.728

Answer

0.482

Answer

0.331

Answer

0.102

Question 22

The Survey of women who are age 18 or older indicated that 86% of married women have one or more children, 31% of never married women have one or more children, and 86% of women who are divorced or widowed have one or more children. The survey also indicated that 48% of women were currently married, 24% had never been married, and 28% were divorced or widowed. If a randomly selected woman has one or more children, what is the probability that she is never married?

Accepted Answer

0.102

Answer

0.567

Answer

0.728

Answer

0.482

Answer

0.331

Question 23

The Survey of women who are age 18 or older indicated that 86% of married women have one or more children, 31% of never married women have one or more children, and 86% of women who are divorced or widowed have one or more children. The survey also indicated that 48% of women were currently married, 24% had never been married, and 28% were divorced or widowed. If a randomly selected woman has one or more children, what is the probability that she is divorced or widowed?

Accepted Answer

0.331

Answer

0.567

Answer

0.728

Answer

0.482

Answer

0.102

Question 24

Given the following statistics, what is the probability that a woman has cancer if she has a positive mammogram result? 1% of women over 50 have breast cancer. 90% of women who have breast cancer test positive on mammograms. 8% of women will have false positives.

Accepted Answer

0.1

Answer

0.31

Answer

0.24

Answer

0.898

Answer

0.012

Question 25

Given the following statistics, what is the probability that a woman has not cancer if she has a positive mammogram result? 1% of women over 50 have breast cancer. 90% of women who have breast cancer test positive on mammograms. 8% of women will have false positives.

Accepted Answer

0.898

Answer

0.1

Answer

0.31

Answer

0.24

Answer

0.012

Question 26

At a certain gas station 40% of the customers request regular gas, 35% request unleaded gas, and 25% request premium gas. Of those customers requesting regular gas, only 30% fill their tanks. Of those customers requesting unleaded gas, 60% fill their tanks, while of those requesting premium, 50% fill their tanks. If the next customer fills the tank, what is the probability that regular gas is requested

Accepted Answer

0.264

Answer

0.45

Answer

0.745

Answer

0.325

Answer

0.61

Question 27

Using the given data, answer the following question. 
What is the probability that a student, taken at random from my class, would have failed the course, given that they failed the final?

Accepted Answer

0.39

Answer

0.72

Answer

0.61

Answer

0.44

Answer

0.58

Question 28

Using the given data, answer the following question.      What is the probability that a student, taken at random from my class, would have passed the course, given that they failed the final?

Accepted Answer

0.61

Answer

0.39

Answer

0.72

Answer

0.44

Answer

0.58

Question 29

Using the given data, answer the following question.
What is the probability that a student, taken at random from my class, would have failed the course, given that they failed the final?

Accepted Answer

0.81

Answer

0.39

Answer

0.72

Answer

0.44

Answer

0.61

Question 30

Using the given data, answer the following question.
What is the probability that a student, taken at random from my class, would have failed the course, given that they failed the final?

Accepted Answer

0.19

Answer

0.39

Answer

0.81

Answer

0.44

Answer

0.61

Question 31

31.	Insurance predictions for probability of auto accident.      Table gives an insurance company’s prediction for the likelihood that a person in a particular age group will have an auto accident during the next year. The company’s policyholders are 20% under the age of 25,   30% between 25 and 39, and 50% over the age of 40. What is the probability that a random policyholder will have an auto accident next year?

Accepted Answer

0.041

Answer

0.145

Answer

0.367

Answer

0.512

Answer

0.845

Question 32

32.	 A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he drives the large car. If he takes the small car, he usually has little trouble parking, and so is at work on time with probability 0.9. If he takes the large car, he is at work on time with probability 0.6. What is the probability that he will not be at work on time tomorrow?

Accepted Answer

7/40

Answer

1/2

Answer

0.6

Answer

33/40

Answer

27/40

Question 33

33.	A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he drives the large car. If he takes the small car, he usually has little trouble parking, and so is at work on time with probability 0.9. If he takes the large car, he is at work on time with probability 0.6. What is the probability that he will be at work on time tomorrow?

Accepted Answer

33/40

Answer

1/2

Answer

0.6

Answer

0.15

Answer

27/40

Question 34

34.	At a certain university, 4% of men are over 6 feet tall and 1% of women are over 6 feet tall. The total student population is divided in the ratio 3:2 in favor of women (the number of women: the number of men). What is the probability that the randomly selected student is over 6 feet tall?

Accepted Answer

0.022

Answer

0.05

Answer

0.14

Answer

0.028

Answer

0.11

Question 35

35.	You enter a chess tournament where your probability of winning a game is 0.3 against half the players, 0.4 against a quarter of the players, and 0.5 against the remaining quarter of the players. You play a game against a randomly chosen opponent. What is the probability of winning?

Accepted Answer

0.375

Answer

1.2

Answer

0.12

Answer

0.028

Answer

0.648

Question 36

36.	If a person has the disease, the test will detect it with probability 0.95. Also, if the person does not have the disease, the test will report that they do not have it with the same probability 0.95. In addition, it is known from previous data that only 1% of the population has this particular disease. What is the probability that a particular person chosen at random will be tested positive?

Accepted Answer

0.059

Answer

0,058

Answer

0,057

Answer

0,056

Answer

0,055

Answer

0,054

Answer

0,060

Question 37

Suppose that you have two bags with white and dark chocolates. Bag 1 has two white chocolates and six dark chocolates. Bag 2 has four white chocolates and two dark chocolates. You choose one bag at random, both being equally likely, and you grab one from the chosen bag. Let A be the event that you grab one white chocolate. Find P(A).

Accepted Answer

11/24

Answer

24/11

Answer

5/13

Answer

2/7

Answer

12/23

Question 38

Amy has two bags of candy. The first bag contains two packs of M&Ms and three packs of Gummi Bears. The second bag contains four packs of M&Ms and two packs of Gummi Bears. Amy chooses the first bag with the probability 0.3 and the second – 0.7. Amy chooses a bag at random and then picks a pack of candy. What is the probability that the pack chosen is M&Ms?

Accepted Answer

44/75

Answer

42/75

Answer

43/75

Answer

45/75

Answer

no woman no cry

Question 39

A fair six-sided die is tossed. You win $2 if the result is a «1», you win $1 if the result is a «6», but otherwise you lose $1. Let X be the amount you win. What is the dispersion of X?

Accepted Answer

1.47

Answer

1.48

Answer

1.49

Answer

Путин

Answer

Навальный

Question 40

40.	Two independent random variables X and Y are given by the following tables of distribution:         Find the mathematical expectation of X+Y?

Accepted Answer

1

Answer

0,5

Answer

0,7

Answer

0,8

Answer

0,1

Question 41

A set of families has the following distribution on number of children:      Determine x_1,x_2, if it is known that M(X)=2,D(X)=1.2?

Accepted Answer

x1=0, x2=1 (1,2 индексы)

Answer

х1=1,х2=0 (1,2 индексы)

Answer

hello

Answer

it's me

Question 42

42.	The lifetime of a machine part has a continuous distribution on the interval (0, 11) with probability density function  f(x) = c(10+x)^-2. Calculate the probability that the lifetime of the machine part is less than 5.

Accepted Answer

7/11

Answer

1/17

Answer

7/20

Answer

10/11

Answer

19/35

Question 43

43.	A random variable X is given by the density function of distribution:    Find the integral function of distribution of the random variable X?

Answer

G

Question 44

44.	A random variable X is given by the integral function of distribution:    Find the mathematical expectation of the random variable X?

Accepted Answer

15/4

Answer

5

Answer

5/2

Answer

3/4

Answer

1

Question 45

45.	A random variable X is given by the integral function of distribution:    Find the mathematical expectation of the random variable X2?

Accepted Answer

1/3

Answer

1

Answer

1/2

Answer

3/4

Question 46

46.	A discrete random variable X is given by the following law of distribution:      By using the Chebyshev inequality estimate the probability that |X – M(X)| > 4.

Accepted Answer

0,18

Answer

1

Answer

0,29

Answer

1/4

Answer

0,35

Question 47

47.	A discrete random variable X is given by the following law of distribution:      By using the Chebyshev inequality estimate the probability that |X – M(X)| > 2.

Accepted Answer

0.74

Answer

0,75

Answer

0,76

Question 48

The probability that a shooter will beat out 10 aces at one shot is equal to 0.1 and the probability to beat out 9 aces is equal to 0.3. Find the probability of the event B – the shooter will beat out no less than 9 aces.

Accepted Answer

0,4

Answer

0,1

Answer

0,3

Answer

0,9

Answer

0,6

Question 49

49.	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.

Accepted Answer

3/5

Answer

3/7

Answer

4/15

Answer

6/15

Answer

1

Question 50

50.	The probability that a shooter will beat out 10 aces at one shot is equal to 0.1 and the probability to beat out 9 aces is equal to 0.3. Find the probability of the event D – the shooter will beat out more than 8 aces.

Accepted Answer

0,4

Answer

0,3

Answer

0,6

Answer

0,5

Question 51

51.	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 – «at least two misses».

Question 52

52.	A random variable X is given by the integral function of distribution:    Find the probability of hit of the random variable X into the interval (1; 5).

Accepted Answer

1

Answer

0,25

Answer

1/3

Answer

2/3

Answer

1/2

Question 53

53.	A random variable X is given by the integral function of distribution:    Find  P ( 5 < X < 9 ).

Accepted Answer

0

Answer

1/4

Answer

3/4

Answer

1/2

Answer

1

Question 54

54.	The probability that a shooter hit in a target at one shot is equal to 0.8. The shooter has made 3 shots. Find the probability of the event – shooter hit in a target exactly one time.

Accepted Answer

0.096

Answer

0.896

Answer

0.512

Answer

0.992

Answer

0.384

Question 55

55.	The probability that a shooter hit in a target at one shot is equal to 0.8. The shooter has made 3 shots. Find the probability of the event – shooter hit in a target exactly three times

Accepted Answer

0.512

Answer

0.896

Answer

0.992

Answer

0.096

Answer

0.384

Question 56

56.	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.

Accepted Answer

0.6

Answer

0.5

Answer

0.9

Answer

0.8

Answer

1

Question 57

57.	The events A, B, C and D form a complete group. The probabilities of the events are those: P(A) = 0,1; P(B) = 0,4; P(C) = 0,1. What is the probability of the event D equal to?

Accepted Answer

0.4

Answer

0.1

Answer

0.3

Answer

0.2

Question 58

58.	A jar of marbles contains 4 blue marbles, 5 red marbles, 1 green marble, and 2 black marbles. A marble is chosen at random from the jar. After returning it again, a second marble is chosen. Choose the correctly calculated probability of events.