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Probability And Statistics For Engineers And Scientists 3rd Edition Anthony Hayter - Solutions

Twenty players compete in a tournament. In how many ways can rankings be assigned to the top five competitors? In how many ways can the best five competitors be chosen (without being in any order)?
There are 17 broken lightbulbs in a box of 100 lightbulbs. A random sample of 3 lightbulbs is chosen without replacement. (a) How many ways are there to choose the sample? (b) How many samples contain no broken lightbulbs? (c) What is the probability that the sample contains no broken
Show that Ckn = Ckn-1 Can you provide an interpretation of this equality?
Which is more likely: obtaining at least one head in two tosses of a fair coin, or at least two heads in four tosses of a fair coin?
Bag 1 contains 6 red balls. 7 blue balls, and 3 green balls. Bag 2 contains 8 red balls, 8 blue balls, and 2 green balls. Bag 3 contains 2 red balls. 9 blue balls, and 8 green balls. Bag 4 contains 4 red balls. 7 blue balls, and no green balls. Bag 1 is chosen with a probability of 0.15, bag 2 with
A fair die is rolled. If an even number is obtained, then that is the recorded score. However, if an odd number is obtained, then a lair coin is tossed. If a head is obtained, then the recorded score is the number on the die, but if a tail is obtained, then the recorded score is twice the number on
How many sequences of length 4 can be made when each component of the sequence can take 5 different values? How many sequences of length 5 can be made when each component of the sequence can take 4 different values? In general, if 3 ≤ n1 ≤ n2, are there more sequences of length n1 with n2
Twenty copying jobs need to be done. If there are four copy machines, in how many ways can five jobs be assigned to each of the four machines? If an additional copier is used, in how many ways can four jobs be assigned to each of the five machines?
A bag contains two counters with each independently equally likely to be either black or white. What is the distribution of X, the number of white counters in the bag? Suppose that a white counter is added to the bag and then one of the three counters is selected at random and taken out of the bag.
It is found that 28% of orders received by a company are from first-time customers, with the other 72% coming from repeal customers. In addition, 75% of the orders from first-time customers are dispatched within one day, and overall 30% of the company's orders are from repeat customers whose orders
When asked to select their favorite opera work, 26% of the respondents selected a piece by Puccini, and 22% of the respondents selected a piece by Verdi. Moreover, 59% of the respondents who selected a piece by Puccini were female, and 45% of the respondents who selected a piece by Verdi were
A random sample of 10 fibers is taken from a collection of 92 fibers that consists of 43 fibers of polymer A, 17 fibers of polymer B, and 32 fibers of polymer C. (a) What is the probability that the sample does not contain any fibers of polymer B? (b) What is the probability that the sample
A lair coin is tossed live times. What is the probability that there is not a sequence of three outcomes of the same kind?
What is the sample space when a winner and a runner-up are chosen in a tournament with four contestants?
Consider telephone calls made to a company's complaint line. Let A be the event that the call is answered within 10 seconds. Let B be the event that the call is answered by one of the company's experienced telephone operators. Let C be the event that the call lasts less than 5 minutes. Let D be the
A manager has 20 different job orders, of which 7 must be assigned to production line I, 7 must be assigned to production line II and 6 must be assigned to production line III. (a) In how many ways can the assignments be made? (b) If the first job and the second job must be assigned to the same
A hand of 3 cards (without replacement) is chosen at random from an ordinary deck of 52 playing cards. (a) What is the probability that the hand contains only diamonds? (b) What is the probability that the hand contains one Ace, one King, and one Queen?
A hand of 4 cards (without replacement) is chosen at random from an ordinary deck of 52 playing cards. (a) What is the probability that the hand does not have any Aces? (b) What is the probability that the hand has exactly one Ace? Suppose now that the 4 cards are taken with replacement. (c) What
Are the following statements true or false? (a) If a fair coin is tossed three times, the probability of obtaining two heads and one tail is the same as the probability of obtaining one head and two tails. (b) If a card is drawn at random from a deck of cards, the probability that it is a heart
There is a probability of 0.55 that a soccer team will win a game. There is also a probability of 0.85 that the soccer team will not have a player sent off in the game. However, if the soccer team does not have a player sent off, then there is a probability of 0.60 that the team will win the game.
A warehouse contains 500 machines. Each machine is either new or used, and each machine has either good quality or bad quality. There are 120 new machines that have bad quality. There are 230 used machines. Suppose that a machine is chosen at random, with each machine being equally likely to be
A class has 250 students. 113 of whom are male, and 167 of whom are mechanical engineers. There are 52 female students who are not mechanical engineers. There are 19 female mechanical engineers who are seniors. (a) If a randomly chosen student is not a mechanical engineer, what is the probability
A business tax form is either filed on time or late, is either from a small or a large business, and is either accurate or inaccurate. There is an 1195 probability that a form is from a small business and is accurate and on time. There is a 13% probability that a form is from a small business and
(a) If four cards are taken at random from a pack of cards without replacement, what is the probability of having exactly two hearts? (b) If four cards are taken at random from a pack of cards without replacement, what is the probability of having exactly two hearts and exactly two clubs? (c) If
A biased coin is known to have a greater probability of recording a head than a tail. How can it be used to determine fairly which team in a football game has the choice of kick-off?
An applicant has a 0.26 probability of passing a test when they take it for the first time, and if they pass it they can move on to the next stage. However, if they fail the test the first time, they must take the test a second time, and when an applicant takes the test for the second time there is
A fair die is rolled live times. What is the probability that the first score is strictly larger than the second score which is strictly larger than the third score which is strictly larger than the fourth score which is strictly larger than the fifth score (i.e., the five scores are strictly
A warning light in the cockpit of a plane is supposed to indicate when a hydraulic pump is inoperative. If the pump is inoperative, then there is a probability of 0.992 that the warning light will come on. However, there is a probability of 0.003 that the warning light will come on even when the
A hand of 10 cards is chosen at random without replacement from a deck of 52 cards. What is the probability that the hand contains exactly 2 Aces, 2 Kings, 3 Queens, and 3 Jacks?
There are 11 items of a product on a shelf in a retail outlet, and unknown to the customers, 4 of the items are overage. Suppose that a customer takes 3 items at random. (a) What is the probability that none of the overage products are selected by the customer? (b) What is the probability that
Among those people who are infected with a certain virus, 32% have strain A, 59% have strain B, and the remaining 9% have strain C. Furthermore. 21% of people infected with strain A of the virus exhibit symptoms, 16% of people infected with strain B of the virus exhibit symptoms, and 63% of people
If two fair dice are thrown, what is the probability that their two scores differ by no more than one?
II a card is chosen at random from a pack of cards, what is the probability of choosing a diamond picture card?
Two cards are drawn from a pack of cards. Is it more likely that two hearts will be drawn when the drawing is with replacement or without replacement?
Two fair dice are thrown. A is the event that the sum of the scores is no larger than 4, and B is the event that the two scores are identical. Calculate the probabilities: (a) A ∩ B (b) A ∪ B (c) Aʹ ∪ B
Two fair dice are thrown, one red and one blue. Calculate: (a) (red die is 5 | sum of scores is 8) (b) (either die is 5 | sum of scores is 8) (c) (sum of scores is 8 | either die is 5)
Consider the network shown in Figure 1.75 with five switches. Suppose that the switches operate independently and that each switch allows a message through with a probability of 0.85. What is the probability that a message will find a route through the network?
An office has four copying machines, and the random variable X measures how many of them are in use at a particular moment in time. Suppose that P(X =0) =0.08. P(X = 1) = 0.11, P(X = 2) = 0.27, and P(X = 3) = 0.33. (a) What is P(X = 4)? (b) Construct and plot the cumulative distribution function.
Suppose that a random variable X can take the value 1, 2, or any other positive integer. Is it possible that P(X = i) = c / i2 for some value of the constant c? Is it possible that P(X = i) = c / i for some value of c?
A consultant has six appointment times that are open, three on Monday and three on Tuesday. Suppose that when making an appointment a client randomly chooses one of the remaining open times, with each of those open times equally likely to be chosen. Let the random variable X be the total number of
Figure 2.18 presents the cumulative distribution function of a random variable. Make a table and line graph of its probability mass function.
Suppose that two fair dice are rolled and that the two numbers recorded are multiplied to obtain a final score. Construct and plot the probability mass function and the cumulative distribution function of the final score.
Two cards are drawn at random from a pack of cards with replacement. Let the random variable .V be the number of cards drawn from the heart suit. (a) Construct the probability mass function. (b) Construct the cumulative distribution function. (c) What is the most likely value of the random variable
Two fair dice, one red and one blue, are rolled. A score is calculated to be twice the value of the blue die if the red die has an even value, and to be the value of the red die minus the value of the blue die if the red die has an odd value. Construct and plot the probability mass function and the
A fair coin is tossed three times. A player wins $1 if the first toss is a head, but loses $1 if the first toss is a tail. Similarly, the player wins $2 if the second toss is a head, but loses $2 if the second toss is a tail, and wins or loses $3 according to the result of the third toss. Let the
Consider Example 5 and the probability values given in Figure 1.38. The company has decided that each television set should be given a quality score calculated in the following manner. A perfect picture scores 4, a good picture scores 2, a satisfactory picture scores 1, and a failed picture scores
Your cards are labeled $1, $2, $3, and $6. A player pays $4, selects two cards at random, and then receives the sum of the winnings indicated on the two cards. Calculate the probability mass function and the cumulative distribution function of the net winnings (that is. winnings minus the $4
A company has five warehouses, only two of which have a particular product in stock. A salesperson calls the five warehouses in a random order until a warehouse with the product is reached. Let the random variable X be the number of calls made by the salesperson, and calculate its probability mass
Consider a random variable measuring the following quantities. In each case state with reasons whether you think it more appropriate to define the random variable as discrete or as continuous. (a) A person's height (b) A student's course grade (c) The thickness of a metal plate (d) The purity of a
The resistance X of an electrical component has a probability density function f (x) = Ax (130 -x2)for resistance values in the range 10 ≤ x ≤ 11.(a) Calculate the value of the constant A.(b) Calculate the cumulative distribution function.(c) What is the probability that the electrical
A random variable X takes values between 4 and 6 with a probability density functionfor 4 ‰¤ x (a) Cheek that the total area under the probability density function is equal to 1.(b) What is P(4.5 ‰¤ X ‰¤ 5.5)?(c) Construct and sketch the cumulative distribution function.(This problem is
A random variable X takes values between -2 and 3 with a probability density functionand f (x) = 0 elsewhere.(a) Find the value of c and sketch the probability density function(b) What is P (-1 ‰¤ X ‰¤ 1 )?(c) Construct and sketch the cumulative distribution function.
A random variable X takes values between 0 and 4 with a cumulative distribution functionfor 0 ‰¤ x ‰¤ 4. (a) What is P(X ‰¤ 2)? (b) What is P(1 ‰¤ X ‰¤ 3)? (c) Construct and sketch the probability density function. (This problem is
A random variable X takes values between 0 and ∞ with a cumulative distribution function F(x) = A + Be-x for 0 ≤ x ≤ ∞. (a) Find the values of A and B sketch the cumulative distribution function. (b) What is P(2 ≤ X ≤ 3)? (c) Construct and sketch the probability density function.
A car panel is spray-painted by a machine, and the technicians are particularly interested in the thickness of the resulting paint layer. Suppose that the random variable A measures the thickness of the paint in millimeters at a randomly chosen point on a randomly chosen car panel, and that A"
Suppose that the random variable V is the time taken by a garage to service a ear. These times are distributed between 0 and 10 hours with a cumulative distribution function F(x) = 4 + B ln(3x + 2) For 0 ≤ x ≤ 10. (a) Find the values of A and B and sketch the cumulative distribution
The bending capabilities of plastic sheets are investigated by bending sheets at increasingly large angles until a deformity appears in the sheet. The angle θ at which the deformity first appears is then recorded. Suppose that this angle takes values between 0o and 10o with a probability density
An archer shoots an arrow at a circular target with a radius of 50 cm. If the arrow hits the target, the distance r between the point of impact and the center of the target is measured. Suppose that this distance has a cumulative distribution functionfor 0 ‰¤ r ‰¤ 50. (a)
Consider again the four copying machines discussed in Problem 2.1.1. What is the expected number of copying machines in use at a particular moment in time? Problem 2.1.1 An office has four copying machines, and the random variable X measures how many of them are in use at a particular moment in
Consider again the random variable described in Problem 2.2.2 with a probability density function offor 4 ‰¤ x ‰¤ 6 and f(x) =0 elsewhere. (a) What is the expected value of this random variable? (b) What is the median of this random variable?
Consider again the random variable described in Problem 2.2.4 with a cumulative distribution function offor 0 ‰¤ x ‰¤ 4. (a) What is the expected value of this random variable? (b) What is the median of this random variable?
Consider again the car panel painting machine discussed in Problem 2.2.6. What is the expected paint thickness? What is the median paint thickness? Problem 2.2.6 A car panel is spray-painted by a machine, and the technicians are particularly interested in the thickness of the resulting paint layer.
Consider again the plastic bending capabilities discussed in Problem 2.2.8. What is the expected deformity angle? What is the median deformity angle? Problem 2.2.8 The bending capabilities of plastic sheets are investigated by bending sheets at increasingly large angles until a deformity appears in
Consider again the archery problem discussed in Problem 2.2.9. What is the expected deviation from the center of the target? What is the median deviation?Problem 2.2.9An archer shoots an arrow at a circular target with a radius of 50 cm. If the arrow hits the target, the distance r between the
Prove that a continuous random variable with a probability density function that is symmetric about a point n has an expected value equal to the point of symmetry μ.
Recall Problem 2.1.1 I concerning the scheduling of appointments with a consultant. What is the expected value of the total number of appointments that have already been made over both days at the moment when Monday's schedule has just been completely tilled.' Problem 2.1.1 The resistance X of an
Recall Problem 2.2.11 concerning the resistance of an electrical component.(a) What is the expected value of the resistance?(b) What is the median value of the resistance?Problem 2.2.11The resistance X of an electrical component has a probability density function f (x) = Ax (130 -x2) for resistance
A random variable has a probability density function f(x) = A(x - 1.5) over the state space 2 ≤ x ≤ 3. (a) What is the value of A? (b) What is the median of the random variable?
Consider again Problem 2.1.3 where the numbers obtained on two fair dice are multiplied to obtain a final score. What is the expected value of this score? Problem 2.1.3 Suppose that two fair dice are rolled and that the two numbers recorded are multiplied to obtain a final score. Construct and plot
Consider again Problem 2.1.4 where two cards are drawn from a pack of cards. Is the expected number of hearts drawn larger when the second drawing is made with or without replacement? Does this answer surprise you? Problem 2.1.4 Two cards are drawn at random from a pack of cards with replacement.
Suppose that a player draws a card at random from a pack of cards, and wins $15 if an Ace, king, Queen, or Jack is obtained, and otherwise wins the face value of the card in dollars. What is the expected amount won by the player? Would you pay $9 to play this game?
Two fair dice, one red and one blue, are rolled, and a fair coin is tossed. If a head is obtained on the coin toss, then a player wins the sum of the scores on the two dice. It a tail is obtained on the coin toss, then the player wins the score on the red die. What are the expected winnings?
A player pays $ 1 to play a game where three fair dice are rolled. It three 6s are obtained the player wins $500. and otherwise the player wins nothing. What are the expected net winnings of this game? Would you want to play this game? Does your answer depend upon how many times you can play the
Suppose that you are organizing the game described at the end of Section 2.3.1. where you charge players $2 to roll two dice, and then you pay them the difference in the scores, [f you fix the dice so that each die has a probability of 0.2 of scoring a 3 and equal probabilities of 0.16 of scoring
Suppose that the random variable X takes the values -2, 1, 4, and 6 with probability values I /3, 1 /6, 1 /3, and 1 /6, respectively. (a) Find the expectation of X. (b) Find the variance of V using the formula Var(X) = E((X - E(X))2) Find the variance of X using the formula Var(X) = E(X2) - (E(X))2
The time taken to serve a customer at a fast-food restaurant has a mean of 75.0 seconds and a standard deviation of 7.3 seconds. Use Chebyshev's inequality to calculate time intervals that have 75% and 89% probabilities of containing a particular service lime.
A machine produces iron bars whose lengths have a mean of 110.8 em and a standard deviation of 0.5 cm. Use Chebyshev*s inequality to obtain a lower bound on the probability that an iron bar chosen at random has a length between 109.55 cm and 112.05 em.
Recall Problems 2.1.11 and 2.3.16 concerning the scheduling of appointments with a consultant. What is the standard deviation of the total number of appointments thai have already been made over both days at the moment when Monday's schedule has just been completely filled?
Recall Problems 2.2.11 and 2.3.17 concerning the resistance of an electrical component. (a) What is the standard deviation of the resistance? (b) What is the 80th percentile of the resistance? What is the 10th percentile of the resistance?
A continuous random variable has a probability density function f(x) = Ax25 for 2 < x < 3. (a) What is the value of A? (b) What is the expectation of the random variable? (c) What is the standard deviation of the random variable? (d) What is the median of the random variable?
In a game a player either loses $1 with a probability 0.25. wins $1 with a probability 0.4. or wins $4 with a probability 0.35. What are the expectation and the standard deviation of the winnings'.'
A random variable X has a probability density function f(x) = A/√x for 3 < x < 4. (a) What is the value of A? (b) What is the cumulative distribution function of X? (c) What is the expected value of X? (d) What is the standard deviation of X? (e) What is the median of X? (f) What is the upper
When a construction project is opened for bidding, two proposals are received with probability 0.11, three proposals are received with probability 0.19, four proposals are received with probability 0.55, and five proposals are received with probability 0.15. (a) What is the expectation of the
A random variable X has a distribution given by the probability density function f(X) = (1 - x)/2 with a state space - 1 < x < 1. (a) What is the expected value of X? (b) What is the standard deviation of X? (c) What is the upper quartile of X?
Consider again the four copying machines discussed in Problems 2.1.1 and 2.3.1. Calculate the variance and standard deviation of the number of copying machines in use at a particular moment.
Consider again the salesperson discussed in Problems 2.1.9 and 2.3.4 who is trying to locate a particular product. Calculate the variance and standard deviation of the number of warehouses called by the salesperson.
Suppose that you are organizing the game described at the end of Section 2.3.1, where you charge players $2 to roll 7 two dice and then you pay them the difference in the scores. What is the variance in your profit from each game? If you are playing a game in which you have positive expected
Consider again the random variable described in Problems 2.2.2 and 2.3.10 with a probability density function of F(x) = 1/x In (1.5) for 4 < x < 6 and f(x) = 0 elsewhere. (a) What is the variance of this random variable. (b) What is the standard deviation of this random variable. (c) Find the upper
Consider again the random variable described in problems 2.2.4 and 2.3.11 with a cumulative distribution function of F(x) = x2/16 for 0 < v < 4. (a) What is the variance of this random variable? (b) What is the standard deviation of this random variable? (c) Find the upper and lower quartiles of
Consider again the ear panel painting machine discussed in Problems 2.2.6 and 2.3.12. (a) What is the variance of the paint thickness'.' (b) What is the standard deviation of the paint thickness? (c) Find the upper and lower quartiles of the paint thickness. (d) What is the interquartile range?
Consider again the plastic bending capabilities discussed in Problems 2.2.8 and 2.3.13. (a) What is the variance of the deformity angle? (b) What is the standard deviation of the deformity angle? (c) Find the upper and lower quartiles of the deformity angle. (d) What is the interquartile range?
Consider again the archery problem discussed in Problems 2.2.9 and 2.3.14. (a) What is the variance of the deviation from the center of the target? (b) What is the standard deviation of the deviation from the center of the target? (c) Find the upper and lower quartiles of the deviation from the
Consider Example 20 on mineral deposits. (a) Show that P(0.8 < X < 1,25 < Y < 30) = 0.092. (b) Show that the iron content has an expected value of 27.36 and a standard deviation of 4.27. (c) Show that conditional on X = 0.55, the iron content has an expected value of 27.14 and a standard deviation
Joint probability distributions of three or more random variables can be interpreted by extending the ideas in this section. I-'or example, suppose that three continuous random variables X, Y, and Z have a joint probability density functionf(x, y, z) = 3xyz2/32for 0 (a) Establish that this is a
Consider Example 19 on air conditioner maintenance. (a) Suppose that a location has only one air conditioner that needs servicing. What is the conditional probability mass function of the service time required, and the conditional expectation and standard deviation of the service time? (b) Suppose
Suppose that two continuous random variables X and Y have a joint probability density function f(x, y) = A(x-3)y for -2 < x < 3 and 4 < y < 6, and f(x, y) = 0 elsewhere. (a) What is the value of A? (b) What is P(0 < X < 1,4 < Y < 5)? (c) Construct the marginal probability density functions fX(x)
A fair coin is tossed four times, and the random variable X is the number of heads in the first three tosses and the random variable Y is the number of heads in the last three tosses. (a) What is the joint probability mass function of X and Y? (b) What are the marginal probability mass functions of
Suppose that two continuous random variables X and Y have a joint probability density function F(x, y) = A(ex+y + e2x-y) for 1 < x < 2 and 0 < y < 3, and f(x, y) = 0 elsewhere. (a) What is the value of A? (b) What is P(l.5 < X < 2, 1 < Y < 2)? (c) Construct the marginal probability density
Two cards are drawn without replacement from a pack of cards, and the random variable X measures the number of hearts drawn and the random variable Y measures the number of clubs drawn. (a) What is the joint probability mass function of X and Y? (b) What are the marginal probability mass functions
Repeat Problem 2.5.6 when the second card is drawn with replacement. In Problem 2.5.6 Two cards are drawn without replacement from a pack of cards, and the random variable X measures the number of hearts drawn and the random variable Y measures the number of clubs drawn. (a) What is the joint

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