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mathematics
calculus

Questions and Answers of Calculus

A retailer purchases 100 of a new brand of DVD player, of which 2 are defective. The purchase agreement says that if he tests 5 chosen at random and finds 1 or more defective, he receives all the DVD
A box containing 500 smart phones has 20 defective phones. If 10 phones are selected together from the box, what is the probability that (a) All 10 are defective? (b) None is defective? (c) Half of
To see whether a bank has enough minority construction company loans, a social agency selects 30 loans to construction companies at random and finds that 2 of them are loans to minority companies. If
A high school principal must select 12 girls at random from the freshman class to serve as hostesses at the junior-senior prom. There are 200 freshman girls, including 20 from minorities, and the
Suppose that an employer plans to hire four people from a group of nine equally qualified people, of whom three are minority candidates. If the employer does not know which candidates are minority
1. Four men and three women are semifinalists in a lottery. From this group, three finalists are to be selected by a drawing. What is the probability that all three finalists will be men? 2. Suppose
A car dealer has 12 different cars that he would like to display, but he has room to display only 5. (a) In how many ways can he pick 5 cars to display? (b) Suppose 8 of the cars are the same color,
Suppose that two openings on an appellate court bench are to be filled from current municipal court judges. The municipal court judges consist of 23 men and 4 women. Find the probability(a) That both
Suppose that an indecisive company owner has ranked the three top officers of his company at random but claims that they earned their jobs because of ability. (a) What is the probability that the
Suppose that 10 computer chips are drawn from a box containing 12 good chips and 4 defective chips. What is the probability that (a) Exactly 4 of the chips are defective? (b) All 10 of the chips are
In New York State there is a state lottery game called Pick 10. The state chooses 20 numbers from 80 numbers, and each player chooses 10 numbers from the same 80 numbers. If all 10 of a player's
A poll asks voters to rank Social Security, the economy, the war on terror, health care, and education in the order of importance. (a) How many rankings are possible? (b) What is the probability that
A task force studying age discrimination wishes to establish a subcommittee. If the subcommittee is to consist of 11 members chosen from a group of 23 men and 19 women, find the probability that the
An insurance company receives 25 claims in a certain month. Company policy mandates that a random selection of 5 of these claims be thoroughly investigated. If 2 of the 25 claims are fraudulent, find
Employees of a firm receive annual reviews. In a certain department, 4 employees received excellent ratings, 15 received good ratings, and 1 received a marginal rating. If 3 employees in this
What is the probability of being dealt a poker hand of 5 cards in which (a) All 5 cards are spades? (b) 4 of the 5 cards are hearts?
A standard deck of cards contains 13 spades, 13 hearts, 13 diamonds, and 13 clubs. A bridge hand contains 13 cards selected at random from the 52-card deck. Find the probability that a hand
A flush (5 cards from the same suit) is an excellent hand in poker. If there are 13 cards in each of four suits in a deck used in the game and there are 5 cards in a hand, what is the probability
Find the probability of being dealt a hand containing 1 spade, 6 hearts, 4 diamonds, and 2 clubs. Assume that a normal deck of 52 cards is used.
On April 24, 1980, the winning Pennsylvania daily number was 666. It was later discovered that on that day the drawing had been fixed by injecting a liquid into all balls except the 4 and the 6,
Two men and a woman are lined up to have their picture taken. If they are arranged at random, what is the probability that (a) The woman will be on the left in the picture? (b) The woman will be in
Suppose that all license plates in a state have three letters and three digits. If a plate is chosen at random, what is the probability that all three letters and all three numbers on the plate will
(a) An automatic teller machine requires that each customer enter a four-digit personal identification number (PIN) when he or she inserts a bank card. If a person finds a bank card and guesses at a
Keys for older General Motors cars had six parts, with three patterns for each part. (a) How many different key designs are possible for these cars? (b) If you find an older GM key and own an older
If the first digit of a seven-digit telephone number cannot be a 0 or a 1, what is the probability that a number chosen at random will have all seven digits the same?
Can the vectors probability vectors? If not, why?1.2. 3. 4.
Transition matrix B, initial probability vector [0.6 0.4].Use the given transition matrix and the initial probability vector to find the second probability vector (two steps after the initial
Transition matrix C, initial probability vector [0.1 0.3 0.6].Use the given transition matrix and the initial probability vector to find the second probability vector (two steps after the initial
Transition matrix D, initial probability vector [0.9 0 0.1].Use the given transition matrix and the initial probability vector to find the second probability vector (two steps after the initial
Use the matrix and initial probability vector in Problem 9 and find the resulting 10th probability vector.
Use the matrix and initial probability vector in Problem 10 and find the resulting 18th probability vector.
Use the matrix and initial probability vector in Problem 11 and find the resulting eighth probability vector.
Use the matrix and initial probability vector in Problem 12 and find the resulting ninth probability vector.
Find the steady-state vector associated with the given transition matrix from Problems 9-12.1. A2. B
Using the given transition matrix and assuming the initial probability vector is [1 0 0], find the probability vectors for the next four steps of the Markov chain. (This initial probability vector
Using the given transition matrix and assuming that a Republican is certain to win the initial election, find the probability vectors for the next four steps of the Markov chain.In a certain city the
If a woman attends church regularly and has one daughter who in turn has one daughter, what is the probability that the granddaughter attends church regularly? The probability that daughters of a
If a woman does not attend church, what is the probability that her granddaughter attends church regularly? The probability that daughters of a mother who attends church regularly will also attend
What is the steady-state vector for this information? The probability that daughters of a mother who attends church regularly will also attend church regularly is 0.8, whereas the probability that
If he drove the Ford today, what are the probabilities for the cars that he will drive 2 days from today? A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two
If he drove the Ford today, what are the probabilities for the cars that he will drive 4 days from today? A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two
If he drove the Audi today, what are the probabilities for the cars that he will drive 4 days from now? A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two
What is the steady-state vector for this problem? A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two days in a row. These are the probabilities that he drives
Suppose a government study estimated that the probability of successive generations of a rural family remaining in a rural area was 0.7 and the probability of successive generations of an urban
A psychologist found that a laboratory mouse placed in a T-maze will go down the same branch with probability 0.8 if there was food there on the previous trial and that it will go down the same
The local community service funding organization in a certain county has classified the population into those who did not give the previous year, those who gave less than their "fair share," and
If A runs the advertising campaign for 3 weeks, find the probability of a customer patronizing each business in the third week.A local business A has two competitors, B and C. No customer patronizes
Find the steady-state vector for this market-that is, the long-range share of the market that each business can expect if the transition matrix holds.A local business A has two competitors, B and C.
If the initial occurrence of those genotypes in the population is 0.5 for DD, 0.4 for Dr, and 0.1 for rr, find the distribution (probability) of each type after the first and second generations.For
1. Find the steady-state vector for this genotype.2. To which generation(s) does the steady-state vector correspond? This is called the Hardy-Weinberg model.For species that reproduce sexually,
Can the matrices be transition matrices? If not, why?1.2. 3. 4.
Transition matrix A, initial probability vector [0.2 0.8].Use the given transition matrix and the initial probability vector to find the second probability vector (two steps after the initial
If one ball is drawn from a bag containing nine balls numbered 1 through 9, what is the probability that the ball's number is (a) Odd? (b) Divisible by 3? (c) Odd and divisible by 3?
A card is drawn at random from a deck of playing cards. What is the probability that it is a king or a red card?
A bag contains 4 red balls numbered 1, 2, 3, 4 and 5 white balls numbered 5, 6, 7, 8, 9. A ball is drawn. What is the probability that the ball (a) Is red and even-numbered? (b) Is red or
A bag contains 4 red balls and 3 black balls. Two balls are drawn at random from the bag without replacement. Find the probability that both balls are red.
A box contains 2 red balls and 3 black balls. Two balls are drawn from the box without replacement. Find the probability that the second ball is red, given that the first ball is black.
A bag contains 8 white balls, 5 red balls, and 7 black balls. If three balls are drawn at random from the bag, with replacement, what is the probability that the first two balls are red and the third
A bag contains 8 white balls, 5 red balls, and 7 black balls. If three balls are drawn at random from the bag, without replacement, what is the probability that the first two balls are red and the
A bag contains 3 red, 5 black, and 8 white balls. Three balls are drawn, without replacement. What is the probability that one of each color is drawn?
An urn contains 4 red and 6 white balls. One ball is drawn, it is not replaced, and a second ball is drawn. What is the probability that one ball is white and one is red?
Urn I contains 3 red and 4 white balls and urn II contains 5 red and 2 white balls. An urn is selected and a ball is drawn. (a) What is the probability that urn I is selected and a red ball is
Bag A contains 3 red, 6 black, and 5 white balls, and bag B contains 4 red, 5 black, and 7 white balls. A bag is selected at random and a ball is drawn. If the ball is white, what is the probability
1. How many 3-letter sets of initials are possible?2. How many combinations of 8 things taken 5 at a time are possible?3. Explain why each given matrix is not a transition matrix for a Markov
For the initial probability vector [0.1 0.9] and the transition matrix A, find the next two probability vectors.
Find the steady-state vector associated with the transition matrix.
1. Senior citizens In a certain city, 30,000 citizens out of 80,000 are over 50 years of age. What is the probability that a citizen selected at random will be 50 years old or younger? 2. Newspapers
Of 100 job applicants to the United Nations, 30 speak French, 40 speak German, and 12 speak both French and German. If an applicant is chosen at random, what is the probability that the applicant
If the probability that an event E occurs is 3 7, what are the odds that (a) E will occur? (b) E will not occur?
1. Find the probability that a person chosen at random has a democratic style and medium productivity.2. Find the probability that an individual chosen at random has high productivity or an
A product must pass an initial inspection, where the probability that it will be rejected is 0.2. If it passes this inspection, it must also pass a second inspection where the probability that it
A regional survey found that 70% of all families who indicated an intention to buy a new car bought a new car within 3 months, that 10% of families who did not indicate an intention to buy a new car
1. A personnel director ranks 4 applicants for a job. How many rankings are possible? 2. An organization wants to select a president, vice president, secretary, and treasurer. If 8 people are willing
Suppose that a fair coin is tossed two times. Construct an equiprobable sample space for the experiment and determine each of the following probabilities. (a) Pr(0 heads) (b) Pr(1 head) (c) Pr(2
A jury can be deadlocked if one person disagrees with the rest. There are 12 ways in which a jury can be deadlocked if one person disagrees, because any one of the 12 jurors could disagree. In how
The RBC Heritage Classic Foundation license plate is available to anyone with a vehicle registered in the state of South Carolina. It costs $75 every two years with $66 to $71 going to the foundation
In a book describing the mass murder of 18 people in northern California, a policewoman was quoted as stating that there are blood groups O, A, B, and AB, positive and negative, blood types M, N, MN,
1. Scheduling A college registrar adds 4 new courses to the list of offerings for the spring semester. If he added the course names in random order at the end of the list, what is the probability
If a person "boxes" (see Problem 44) a 4-digit number with 4 different digits in Pennsylvania's Pick Four Lottery, what is the probability that the person will win?
A supplier has 200 compact discs, of which 10% are known to be defective. If a music store purchases 10 of these discs, what is the probability that (a) None of the discs is defective? (b) 2 of the
A sample of 6 fuses is drawn from a lot containing 10 good fuses and 2 defective fuses. Find the probability that the number of defective fuses is (a) Exactly 1. (b) At least 1.
If the initial distribution probabilities of families in a certain population are [0.7 0.2 0.1], find the distribution for the next three generations.Suppose people in a certain community are
Suppose that a fair coin is tossed three times. Construct an equiprobable sample space for the experiment and determine each of the following probabilities. (a) Pr(2 heads) (b) Pr(3 heads) (c) Pr(1
Find the expected long-term probability distribution of families by income level.Suppose people in a certain community are classified as being low-income (L), middle-income (M), or high-income (H).
1. A card is drawn at random from an ordinary deck of 52 playing cards. What is the probability that it is a queen or a jack? 2. A deck of 52 cards is shuffled. A card is drawn, it is replaced, the
Suppose a fair die is rolled 18 times. (a) What is the probability p that a 4 will occur each time the die is rolled? (b) What is the probability q that a 4 will not occur each time the die is
A bag contains 6 red balls and 4 black balls. If we draw 5 balls, with each one replaced before the next is drawn, what is the probability that at least 2 balls drawn will be red?
Suppose a pair of dice is thrown 4 times. What is the probability that a sum of 9 occurs exactly 2 times?
If a pair of dice is thrown 4 times, find the probability that a sum of 7 occurs at least 3 times.
Suppose the probability that a marksman will hit a target each time he shoots is 0.85. If he fires 10 shots at a target, what is the probability he will hit it 8 times?
A family has 4 children. If the probability that each child is a girl is 0.5, what is the probability that (a) Half of the children are girls? (b) All of the children are girls?
A multiple-choice test has 10 questions and 5 choices for each question. If a student is totally unprepared and guesses on each question, what is the probability that (a) She will answer every
The manager of a store buys weather radios in lots of 48. Suppose that, on the average, 4 out of each group of 48 are defective. The manager randomly selects 4 radios out of the group to test. (a)
A hospital buys thermometers in lots of 1000. On the average, one out of 1000 is defective. If 10 are selected from one lot, what is the probability that none is defective?
The probability that a certain couple will have a blue-eyed child is 1 4, and they have 4 children. What is the probability that (a) 1 of their children has blue eyes? (b) 2 of their children have
Suppose a fair coin is tossed 12 times. (a) What is the probability p that a head will occur each time the coin is tossed? (b) What is the probability q that a head will not occur each time the coin
If the probability that a certain couple will have a blue-eyed child is 1/2, and they have 4 children, what is the probability that (a) None of the children has blue eyes? (b) At least 1 child has
Suppose that 10% of the patients who have a certain disease die from it. If 5 patients have the disease, what is the probability that (a) Exactly 2 patients will die from it? (b) No patients will die
If the ratio of boys born to girls born is 105 to 100, and if 6 children are born in a certain hospital in a day, what is the probability that 4 of them are boys?
If records indicate that 4 houses out of 1000 are expected to be damaged by fire in any year, what is the probability that a woman who owns 10 houses will have fire damage in 2 of them in a year?

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