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Introduction To Probability 1st Edition Mark Daniel Ward, Ellen Gundlach - Solutions
Let X denotes the number of calories that a person eats in a single day. Suppose that X is normally distributed with μx = 2000 and σ2X = 10,000. If the person's eating habits are independent from day to day, find the probability that the person eats 735,000 calories or more during a 365-day year.
At a local store, there are 14 customers during a given evening. Each customer has an expected purchase of I $4.90, with a standard deviation of $1.50. If the customers' purchases are assumed to be independent and approximately Normal, what is the approximate probability that the revenue exceeds
At the library, a student checks out 10 books. She estimates that each book has a normally distributed weight, with mean 12 ounces and standard deviation of 3 ounces. What is the probability that the total weight of the books exceeds 125 ounces altogether?
A student has 23 candy sticks in a bag, with lengths that are normally distributed. Each stick is, on average 1.8 cm long, with standard deviation 0.5 cm. What is the probability that the total length of the candy is less than 40 cm?
According to www.fowlerfarms, com/apple_a_day.htm, a certain kind of apple weighs 150 grams, on average. Suppose that the standard deviation of this type of apple is 20 grams. When picking 66 such apples, what is the probability that they weigh (altogether) 9966 grams or more?
At a certain university, each student who tries to purchase concert tickets for an upcoming show is successful at connecting to the concert ticket website with probability 0.85. If unsuccessful in logging on, he or she tries again a few minutes later, over and over, until finally getting tickets.
Each morning a student takes a shower that lasts 15 minutes, with a standard deviation of 4 minutes. Find an estimate of the probability that the student spends between 11 and 12 hours in the shower during a 45-day period.
The registrar estimates that each student enrolls for 15.3 credits on average, with a standard deviation of 1.2 credits per student. Estimate the probability that, on a campus with 4000 enrolled students, the total number of enrolled credit hours is strictly between 61,100 and 61,300.
A student builds a toy geyser for her engineering class that has height (in inches) of density fx(x) = 1/8 e-x/8 for x > 0, and fx(x) = 0 otherwise. If she makes 20 such geysers, what is the rough probability that the average geyser height, among these 20 geysers, is 7 inches or more?
A group of students in a probability course is skeptical about Geometric random variables, so one day; they all bring a die to class. Each of the 40 students rolls her/his die repeatedly. Each student stops upon their first 5 that appears, independent of the other students. What is the estimated
Suppose that each call a student makes to his girlfriend has an average length of 12 minutes, and the standard deviation of the length of each call is also 3 minutes. Estimate the probability that he completely uses up a 300-minute calling card while talking to his girlfriend during 24
Suppose that the number of pages in an issue of a journal is uniformly distributed between 256 and 384. During 40 years of publication, the journal publishes a total of 160 issues. Give an estimate for the probability that between 51,000 and 52,000 pages (inclusive) were used for these 160 issues.
A car dealership expects to sell 6 cars per day, with standard deviation of 2.1 cars per day. During a 100-day period, estimate the probability that they sell strictly more than 625 cars.
If the amount of rainfall in a given region is assumed to be Uniform on the interval [0.2, 4.0] (in inches) each month, what is the approximate probability that there are 53 or more inches of rain during a 24-month period?
In a math class with 200 students, suppose that the students' decisions to attend the class are independent, and each student attends with probability 93%. On a given day, find the approximate probability that strictly fewer than 179 students attend.
Consider a group of students whose are assigned to work a random number of hours. Their hours per student, per week, are modeled by a Binomial random variable with n = 20 and p = 0.8. (Each hour assigned to work will count as a "success" in the Binomial model.) If there are 100 students in the
At an auction, exactly 282 people place requests for an item. The bids are placed "blindly," which means that they are placed independently, without knowledge of the actions of any other bidders. Assume that each bid (measured in dollars) is a Continuous Uniform random variable on the interval
A restaurant claims that each of their burgers has a 1/4 pound of meat after it is prepared. Of course this is not exact. The students in a local high school measured some of these burgers for a science project, and they concluded that the expected weight of such a burger is 0.251 pounds, but the
Consider customers who arrive at a checkout counter with an average rate of 8 per hour following a Poisson distribution. Find the probability that strictly more than 70 customers arrive during the eight-hour shift.
A particular basketball player successfully scores about 75% of his free throw attempts. Approximate the probability that he makes strictly more than 145 of 200 attempts.
If a certain type of slot machine has only a probability of 0.0001 of yielding a jackpot on each game at a certain casino in Vegas, and the patrons play those slots 250,000 times during a given month, estimate the probability that the casino will have 30 or more jackpot rewards to payout during the
An ice cream shop estimates that its number of patrons per day is Poisson with mean 19. What is the estimated probability that they have 20 or more customers on a given day?
In the early testing stages of processor manufacturing, 40% of the processors will fail in some way. If 500 processors are manufactured at this stage, what is a rough estimate for the probability that strictly less than 180 of the processors will fail in some way?
When buying a pack of strawberries at the farmer's market, there is a 0.10 chance that there will be a rotten berry in the pack. If a restaurant buys 300 packs of strawberries for their desserts, what is the approximate probability that fewer than 25 packs will contain a rotten berry?
If 6% of all passengers are screened with two rounds of security at the airport, and southwest has 8 flights with 180 passengers each, what is the approximate probability that 80 or more of these Southwest passengers will receive this extra level of screening?
A bag of microwave popcorn comes with 200 kernels of corn. When micro waved for exactly 3 minutes, an individual kernel pops 90% of the time. What is the approximate probability that there are strictly less than 10 unpopped kernels in such a bag?
Twenty-five students want burritos, but the dining hall only has one burrito maker. Each burrito takes an average of 72.5 seconds to cook, with standard deviation 3.2 seconds.a. Estimate the probability that all twenty-five students can cook their burritos in half an hour or less, if we ignore the
A certain baseball player has, on average, 0.7 RBI's (runs batted in) per game, with standard deviation of 0.2. What is the approximate probability that the player has at most 110 RBI's during a given season, which contains 162 games?
At a certain university, a large course has 1500 registered students; the course can be viewed online if a student does not feel like walking to the lecture hall. Each student decides each day (independently of the other students and independent of her/his own prior behavior) whether to attend the
A person plays a certain lottery game 10,000 times during his life. The chance of winning is 1/5000 during each attempt, and the attempts are independent. Approximate the probability that he wins 2 or more times during his life.
Back in 1970, a television commercial for Tootsie Pops (a candy) asked how many licks would be needed to get to the Tootsie Roll Center of a Tootsie Pop. This question has become somewhat famous and well-known among children. High school, undergraduate, and Ph.D. students have all performed
The number of strawberry milkshakes that are sold at a local diner in a given day has an average of 97.5 per day and standard deviation of 7.8. In a given 30-day month, what is the approximate probability that they sell more than 3000 milkshakes?
A student is learning to spell some difficult words. He estimates that it takes him 6 attempts to learn each word, with a variance of 3.5 per word. Find the approximate probability that he will need 125 or more attempts to learn 20 words.
Water is sold in bottles that are advertised to contain 1-liter each. Barbara is an inspector for the company and notices that the amount of water in each bottle, unfortunately, has an average of only 0.99 liters, and surprisingly high standard deviation of 0.03 liters. Estimate the probability
Rafael is pursuing a major in computer science. He notices that a memory chip containing 212 = 4096 bits is full of data that seems to have been generated, bit-by-bit, at random, with 0's and l equally likely and the bits are stored independently. If the each bit is equally likely to be a 0 or 1,
Josephine estimates that the worms in her composter can eat, on average, 3.05 pounds of waste per week, with standard deviation of 0.3 pounds. She tests this conjecture with her worms over a 50-week period. Estimate the probability that the average over the 50-week period exceeds 3 pounds per week.
A total of 40,000 students on a university campus independently choose whether to go to the dining hall for dinner each day. Each student has dinner there, on a given day, with probability 0.84.a. What is the distribution of the total number of visits by the students to the dining hall during a
Five hundred people participate in a 12-week survey to see how many times that they go to a religious service. Each person's attendance has a Binomial distribution with n = 12 and p = 0.7. Assume that the people behave independently with regard to attendance.a. What is the actual distribution of
It is known that the lifetime of Save-Bucks batteries follows an Exponential distribution with a mean of 40 hours. The lifetime of LittleMoola batteries follows an Exponential distribution with a mean of 30 hours. a. What is the probability a SaveBucks battery will last more than 48 hours? b. What
The power has gone out in your house, and you are using a small flashlight to read your probability book. You have three batteries available to you, all of the same brand, but you can't tell which brand in the dark. The flashlight uses only one battery at a time. Use the information about the two
The SaveBucks Battery Company wants to wait until 60% of the batteries are sold at the corner convenience store before they ship more. After years of study, they have determined that the proportion of batteries which are sold after 6 weeks follows a Beta distribution with a = 2 and (3 = 3. a. What
While walking from the car into your dormitory, you dropped your engagement ring somewhere in the snow. The path is straight and 30 feet long. You are distraught because the density of its location seems to be constant along this 30-foot route. a. What is the probability that the ring is within 12
A lifeguard has to jump in to save a swimmer in trouble at the lake at Possum Park an average of 2 times a week, according to a Poisson distribution.a. What is the probability that he will save more than 25 swimmers in the 10-week season he works?b. If you know he saved exactly 1 swimmer on his
A patient has been diagnosed with a serious disease. The lifetimes of patients with this disease are approximately Exponential with expected value 10 years, after their initial diagnosis. a. What is the probability that a patient who is diagnosed today will still be alive 15 years from now? b.
A nurse specializes in caring for patients diagnosed with the disease described in Exercise 38.20. a. The nurse cares for one patient at a time since the care is given in the patient's home, and constant monitoring of the patient is needed. She begins caring for that patient as soon as the
The fraction of juice which can be squeezed from an apple is a random variable with the following density:a. Find the value of k that makes this a valid probability density function.b. What is the expected fraction of juice which can be squeezed from an apple?c. What is the standard deviation in
If X has a mean 0.5 and a variance 0.25, find the parameters of the distribution (if possible) if X is: a. Binomial b. Exponential c. Normal d. Uniform e. Geometric
If X has a mean 3 and a variance 2.5, find the parameters of the distribution (if possible) if X is: a. Binomial b. Exponential c. Normal d. Uniform e. Geometric
The cable technician guarantees he'll arrive sometime between 8 AM and noon, but he can't be any more specific than that.a. What is the probability that he will come in between 9:30 and 10:45 AM?b. You have a meeting at noon, and you're hoping that the cable technician will come before 11:30 AM so
The amount of time you use on your cell phone each month is approximately normally distributed with a mean of 623 minutes and a standard deviation of 24 minutes.a. What is the probability that you will talk more than 700 minutes (when the extra charges apply) next month?b. What range of talk-times
The arrival times of students at the health center at a university is known to have a Poisson distribution with an average rate of 6 sick students arriving per hour.a. What is the expected number of students who arrive in the next 30 minutes?b. What is the expected time that will elapse between the
An insurance company expects that 10% of its moderate-risk drivers will be involved in an accident during the first 31 days of the year.a. What is the average number of days that you would expect to wait for a moderate-risk driver, chosen at random, to be involved in an accident?b. What portion of
For low-risk drivers, the waiting time for their first accident follows an Exponential distribution with a mean of 7 years. For high-risk drivers, the waiting time for their first accident follows an Exponential distribution with a mean of 2 years. Assuming that low-risk drivers and high-risk
A renter's insurance policy is written to cover a loss, X, where X has a Uniform distribution with boundaries ranging from no loss up to a maximum of $2000.a. What is the expected loss?b. What is the variance in the loss?c. If a deductible is set at $500, what is the probability that the loss would
Three investments in high-tech stocks are thought to have returns which are Exponential random variables with means of 2, 3, and 4. Determine the probability that the maximum return from these 3 investments is more than 5.
The lifetime of a computer component purchased at World Wide Computers is exponentially distributed with a mean of 2 years. a. What is the probability that the component fails in the first year? b. What is the probability that the component fails in the second or third year (between the 1-year mark
Use the information in Exercise 38.32 about the computer components. If the component fails during the first year after purchase, the owner will refund the buyer's purchase price. If the component fails during the second or third year after purchase, he will refund 25% of the buyer's purchase
The time an energy-saving light bulb will last without failing has an Exponential distribution with a median of 3 years. Calculate the probability that a light bulb, selected at random, will last at least 4 years.
Let X be the number of customers who will want to buy a chocolate chip cookie in the next hour if, according to bakery records, an average of 3 customers per hour want to buy a chocolate chip cookie.For this exercise state which continuous or discrete distribution would be most appropriate and why
Let X be the exact arrival time of the first customer if we know that exactly one customer arrived between 8:03 and 8:12 AM.For this exercise state which continuous or discrete distribution would be most appropriate and why you think so. The choices are:
Each day, Amy eats lunch at the cafeteria. She chooses pizza as her main dish with probability 40%, and her behavior each day is independent of all the other days. Let X denotes the number of days she chooses pizza in a 10-day period. Let Y = 10 - X denote the number of days in which Amy does not
Let X and Y be Uniformly distributed on the diamond-shaped region with corners located at (1, 0), (0, 1), (-1,0), and (0,-1).
Lloyd eats X pieces of pizza, where X is an integer-valued random variable that is equally likely to be any of the values 1 through 8, inclusive. He takes the extra pizza home to his family, but if he eats too much pizza, he buys an extra pizza to take home too. Let Y be the number of pieces that
Let X and Y be jointly distributed on the portion of the Cartesian plane between the curves y = ˆšx and y = x2. Let the joint density of X, Y be fX,Y(x, y) =8y/3x in this regions, and fX,Y (x, y) = 0 otherwise. This is a density because
Roll two dice. Let X denotes the maximum value that appears, and let Y denote the minimum value that appears.
Let X be Uniform on the interval [0, 1], and let Y = ex.
Let X be an Exponentially distributed random variable with expected value 1/2. Let Y = X2.
Consider X and Y such that the joint density fX,Y (x,y) of X and Y is Uniform on the square where 0 ≤ X, Y ≤ 1. In other words,fX,Y(x, y) = 1if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1,and fX,Y (x,y) = 0 otherwise.a. Are X and Y dependent or independent?b. Find the covariance Cov(X2, X + Y) of X2 and X
Roll two 4-sided dice (not 6-sided dice). Let X be the minimum value, and let Y be the maximum value. Find the covariance of X and Y.
Let X and Y correspond to the horizontal and vertical coordinates in the triangle with corners at (2,0), (0,2), and the origin. Let fX,Y (x,y) = 15/28(xy2 + y) for (x, y) inside the triangle, and fX,Y (x,y) = 0 otherwise.a. Find the covariance of X and Y.b. Find the correlation of X and Y.
If X and Y have a constant joint density on the triangle where 0 ≤ y ≤ x ≤ 1, compute Cov(X, Y)
Let X be uniformly distributed on the interval [0, 10], and let Y = 10 - X.a. Find the covariance of X and Y.b. Find the correlation of X and Y.
Draw 5 cards, without replacement, from a standard deck of cards. Let X be the number of hearts selected. Find the variance of X.
Suppose that Sasha picks an integer X at random between 1 and 6 (inclusive). Then Ravi picks a different integer Y at random, from the 5 integers that remain. Assume that all (6) (5) = 30 choices are equally likely. Find the correlation p(X, Y) of X and Y.
A total of 3n bears are in a bucket: n are red, n are yellow, and n are blue. A child begins grabbing the bears at random, with all selections equally likely. The bears are selected "without replacement," i.e., she never puts the bears back after she grabs them. Find the variance of the number of
An accountant must meet with one more clients before he can go home. The amount of time X (in minutes) that he meets with the client is uniformly distributed on the interval [40, 60]. The total length of time Y (also in minutes) that he must remain in the office is 1.3X + 10.a. Find the covariance
You are babysitting for two children, Abby and Bill. They have a bucket of 30 crayons, 20 of which are unbroken, and the other 10 are broken. They each choose a crayon, without replacement. Let X = 1 if Abby gets an unbroken crayon, and X = 0 otherwise. Similarly, let Y = 1 if Bill gets an unbroken
Let X be any random variable, and Y = a - X, where "a" is a constant.
Let X be uniformly distributed on the interval [0, π], and let Y = cos X.
Henry and Sally each choose 1 candy from a bag of treats, without replacement. There are 20 sweet candies and 3 sour candies. Let X = 1 if Henry's candy is sweet, or X = 0 otherwise. Let Y = 1 if Sally's candy is sweet, or Y = 0 otherwise.
A grandmother is knitting wool sweaters to donate to a non-profit organization. She knits pink and blue ones for girls and boys, respectively. It takes 3 balls of yarn to knit one. She has promised to donate 10 sweaters, but she has forgotten how many of the recipients are boys and how many are
Let X and Y have a joint Uniform distribution on the triangle with corners at (0,2), (2,0), and the origin.
Wake up times. Suppose that, on a random Sunday, your roommate wakes up at some time uniformly distributed between 12 noon and 3 PM. suppose that you awake at 2 PM that day. If your roommate is not yet awake when you get up at 2 PM, when do you expect him to wake up?
Roll two 6-sided dice. Let X denotes the minimum value that appears, and let Y denote the maximum value that appears. a. Find E(Y | X = 3). b. Find E(X + Y | X = 3).
A child rolls a pair of dice, one of which is blue and one of which is red. a. Given that the sum of the dice is 8, find the probability that the red die shows the value 4. b. Now the child rolls a pair of dice that look the same (i.e., which are not painted). Given that the sum of the dice is 8,
Helen and Joe play guitar together every day at lunchtime. The number of songs that they play on a given day has a Poisson distribution, with an average of 5 songs per day. Regardless of how many songs they will play that day, Helen and Joe always flip a coin at the start of each song, to decide
Let Y denote the value of the dealer's card that can be seen by all the players, in a game of Black Jack. Let X be a Bernoulli random variable that indicates whether the dealer must stay (i.e., not take another card). Given Y = 10, find the expected value of X. Hint: In Black Jack, if the dealer
Let X and Y denote, respectively, the x- and y-coordinates of the location of a random chosen point that is uniformly distributed in a region. Given Y = 1, find the expected value of X when the region is: a. A circle of radius 2, centered at the origin b. A semicircle corresponding to the
Sandra rolls two 10-sided dice. Let Y be the sum of the two dice, and let X be the value on the first die that she rolls. Given Y = 15, what is the expected value of XI
Suppose that Harrison comes to pick up Rosita for a date. Rosita will be ready at a time that is uniformly distributed between 7 PM and 7:10 PM. If Harrison arrives at 7:07 PM, how long does he expect to have to wait until Rosita is ready? (If she is already ready when he arrives, then his waiting
A point is chosen uniformly inside a circle of radius 2, centered at the origin. If we are given that the point lands exactly on the x-axis, find the expected distance of the point to the origin.
Let X and Y correspond to the horizontal and vertical co- ordinates in the triangle with corners at (2,0), (0,2), and the origin. Let fX,Y(x,y) = 15/28(xy2 + y) for (x,y) inside the triangle, and fX,Y (x,y) = 0 otherwise. Find E(X | Y = 1.5).
Consider a man and a woman who arrive at a certain location; whoever arrives first will wait for the other to arrive. If X and Y denote (respectively) the arrival times of the man and the woman after noon, in minutes, assume that X and Y are independent and each Uniformly distributed on [0, 60].
As in Exercise 24 and Example 36.10 assume that the height (in inches) of an American female is Normal with expected value μ = 64 and standard deviation σ1 = 2.5. Also assume that the height of an American male is Normal with expected value μ2 = 69 and standard deviation σ2 = 3.0. A man and a
While Juanita waits for her Mac and Cheese to boil, she works on her homework for 1/2 of the time. She has been waiting for 3 minutes already (so she has already done 1.5 minutes of homework). She doesn't know how much more time is needed. She estimates that the remaining time (in minutes) until
There are 20 pieces of candy in a machine: 5 Hershey kisses, 6 Kit Kats, 4 Snickers, and 5 Paydays. The machine randomly selects one of the remaining candies when someone makes a purchase, and the candies are not replenished after the purchase (i.e., purchases are made without replacement). If X
While waiting for the bus on a snowy morning, the expected waiting time (including unusual delays for snow!), is 12 minutes. a. Use the Markov inequality to find an upper bound on the probability that the bus takes 15 minutes or more to arrive. b. In the same scenario as above, assume that the
A talented soccer player is expected to score 21 goals in a given soccer season. a. Find an upper bound on the probability that he scores 30 goals or more during the season. b. If he has a standard deviation of 6.5 goals during the season, give a bound on the probability that he scores between 8
The flight time of a plane flight from Denver to New York City is has an average flight time of 3 hours and 16 minutes, and standard deviation is 30 minutes. a. Give a bound on the probability that such a flight takes 6 hours or longer. b. Give a bound on the probability that the announced arrival
The average temperature in December is 27°. a. Find a bound on the probability of the temperature falling outside the range -50° to 50°. b. If the standard deviation of the temperature is 7°, then what is the probability that the temperature falls outside the range of 14° to 40°?
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