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Introduction To Probability 1st Edition Mark Daniel Ward, Ellen Gundlach - Solutions

At a certain hot dog stand, during the working day, the number of people who arrive to eat is Poisson, with an average of 1 person every 2 minutes.a. What is the probability that exactly 3 people arrive during the next 10 minutes?b. What is the probability that nobody arrives during the next 10
Approximately 6.85 left-handed people are killed each day by using an object or machinery designed for right-handed people. Let X be the number of left-handed people killed this way in one day.a. What values can X take? Why is this a Poisson situation? What is the parameter?b. What is the
Suppose that, on average, 3 telemarketers call your house during a 7-day period.a. What is the mass of the number of telemarketers calling your house during one day?b. What is the probability that no telemarketers will call your house tomorrow?c. What is the probability that exactly 2 telemarketers
The number of Yankees fans shopping at a sports store in a given hour is Poisson with mean 8. The number of Red Sox fans shopping at the same store is Poisson with mean 6 per hour. Assume that the numbers of fans of the two types are independent. In particular, there is no person who is
Suppose that the number of men who visit a website is Poisson, with mean 12 per minute, and the number of women who visit the same site is also Poisson, with mean 15 per minute. Assume that the number of men and women are independent. a. During the next 10 seconds, what is the probability that 1
At the local stadium, there are 60,000 fans attending a football game. It is well known that only a few people at the game will be impartial (i.e., will not care about the outcome of the game). Suppose that each person at the game has probability 1 / 10,000 of being impartial (and that the
During the holiday rush, there are 100,000 shoppers in a certain region. Each of these shoppers is extremely likely to make a purchase. Suppose that a person makes a purchase with probability 49,999/50,000 and declines to make a purchase with probability 1/50,000. Let X be the number of people who
If it costs the bakery owner $20 to keep the bakery open each hour (staff, electricity, etc.), each customer spends $2.50, and customers arrive at the bakery at an average rate of 6 per half-hour, will the bakery be able to stay in business (i.e., will the bakery make more money than it spends in
Children laugh an average of 16.67 times an hour and adults laugh an average of 0.6667 times an hour. Let X be the number of times a child laughs in an hour, and let Y be the number of times an adult laughs in an hour. a. What is the probability that a child will laugh exactly 20 times in the next
According to the Guinness Book of World Records (2005), the fastest pumpkin-carver on record, Steven Clarke, carved 42 pumpkins an hour. Assume this is his average rate. Let X be the number of pumpkins Steven carves in an hour. (We suppose that the carver can steadily maintain work at his record
The probability of a mother giving birth to quardruplets is 1 in 729,000. An obstetrician checks the records of 1,000,000 mothers, to see how many mothers of quadruplets are in her database.a. Which distribution is technically appropriate in this situation? What are e parameters?b. Write a formula
Roseate spoonbills are common along I the Gulf Coast of the U.S. See nationalzoo.si.edu/Animals/Birds/Facts/fact-rosespoonbill.cfm for more information. Suppose that the number of j roseate spoonbills that fly overhead in 1 hour (at a nature preserve) has a Poisson distribution with mean 2. Also
In a person's lifetime, he enters a local daily raffle 10,000 times, but his chances of winning each time are only 1 in 5000.Approximate the probability that he wins at least one time during his life time.
There is a certain disease in America which is very rare. Each person has a probability of 1 in 100,000,000 of having the disease. Assume that there are 310,000,000 people in America. a. Approximate the probability that nobody in America actually has the disease. b. Approximate the probability that
Cars pass an intersection at an average rate of 4 cars per 5-minute interval. Let X be the number of cars that pass in the next 5 minutes. a. Show the labeled graph of the mass for X. b. Show the labeled graph of the CDF for X.
You are hungry and decide to go to patronize your office's vending machine. There are 6 bags of potato chips, 7 bags of pretzels, and 5 packs of chocolate chip cookies. Unfortunately there is something wrong with the buttons on the vending machine, and it will not let you type in your selection.
I have 6 chocolate chunk granola bars, 10 raspberry granola bars, and 8 chocolate chip granola bars. I grab 3 without looking. a. What is the probability that I grab exactly 2 that are chocolate (either chunk or chip)? b. What is the probability that I grab fewer than 2 that are chocolate? c. What
Henry has 10 pants: 4 dress slacks and 6 jeans. In a hurry while getting ready for a trip, he asks his kids to throw 3 pants in a suitcase for him without specifying which kind of pants he needs. a. What is the probability the kids correctly throw in 2 pairs of dress slacks and 1 pair of jeans? b.
As a prank, your roommate removed all the labels from all the cans in your pantry and shuffled them around. Now you have no idea what is in a can when you get ready to cook dinner! You know i hat you have 8 cans of corn and 5 cans of beans. You don'1 want to waste any of the cans, so the best thing
At a certain restaurant, during the working day, there are 12 customers. Seven of them have pizza, and the other five have burgers. Suppose that a person is conducting a survey of three customers at the restaurant, and he conducts the survey without replacement, i.e., he does not talk to the same
Carlos "Coyote" Jones owns quite a few harmonicas. In particular, he has 7 professional harmonicas and 12 cheaper harmonicas. They have relatively similar shapes, so when he reaches into his harmonica container without looking, he does not notice a difference between them. Suppose that he grabs 8
Given a standard deck of cards, you are dealt a hand of 13. a. What is the probability that more than half the spades in the deck wind up in your hand? b. How many spades are you expected to get in this hand?
You are going on a trip with some friends, and you need to rent 4 large cars to get all of you to the beach. There are 15 SUVs and 10 minivans on the lot. The rental car dealer will pick 4 cars at random. You are hoping to get SUVs because you think they look more stylish.a. What is a success in
Skittles candies come in the colors red, orange, yellow, green, and purple. Green is your favorite color. Assume that the colors are evenly distributed in a bag of 45 candies. You will eat 10 candies, counting the number of green ones as you eat.a. What is a success in this story? What is a
You are picking a bouquet of 20 flowers for your mother at random from a garden with 25 dahlias, 35 daisies, and 42 bachelor's buttons. Your mother likes daisies the best, and you want to end up with 15 of those in your bouquet.a. What is a success here? What is a failure?b. Explain in words what X
Vera wants to buy 20 copies of the DVD of "Bridget Jones' Diary" as a graduation gift for the women in her sorority. She buys them from a large internet company with 50,000 of the DVDs in stock. However 10,000 of the DVDs labeled as "Bridget Jones" were really "Terminator" and just mislabeled,
Dr. I.M.A. Genius has developed an experimental vaccine for an allergy to cat fur. He needs to test it out on some people to make sure it works. He has a list of 4000 people from his clinic, and 1250 of those have an allergy to cat fur. His receptionist randomly calls in 50 different people from
You have 5 freshmen, 6 sophomores, 10 juniors, and 2 seniors in your probability class. The teacher randomly selects 3 students to go to the board to do problems. a. What is the probability all 3 students will be juniors? b. What is the probability all 3 students will be from the same year? c. What
Skittles candies come in 5 different colors: red, orange, yellow, green, and purple. You have a bowl of the candies, so you reach in and grab one. Each of the five candies is equally likely to appear.a. Why is this a Discrete Uniform distribution situation? What is the parameter?b. What is the
A computer is randomly selecting a 4-digit ID num-"r for you, and there are 10,000 possibilities, each equally likely. (0000 would e the smallest possible number, and 9999 would be the biggest.)a. What is the probability your new ID number happens to match your birth date (e.g., 0822 for August
Prove that the expected value of a Discrete Uniform random variable on the set {1,2,...,JV} is E(X) = (N + l) / 2. Also prove that the variance is Var(X) = (N2 - 1) / 12.
Philip and Callum play a game of Chinese checkers. Each time they play a game, Philip has a 0.7 chance of winning. Assume the games are independent.a. In a tournament, Philip and Callum will play 8 games. What is the probability Philip wins exactly 6 games?b. Philip and Callum decide to keep
According to past experience, 20% of customers coming into Frank's car dealership will buy one of his cars. Assume customers are independent and sampled from a very large population. Assume his dealership is open 7 days a week.a. What is the probability the next person who walks in the door will
On average, 9 babies are born per 24-hour day at the local hospital.a. What is the probability that at least one baby will be born today on the 8 am to 4 pm shift?b. What is the probability that at least one baby will be born in the next hour?c. What is the probability that exactly 4 babies will be
There are 5 juniors and 10 seniors (one of which is Amelia), trying to win a scholarship to a summer music program. Only 3 students can win, and the winners will be selected randomly by pulling names out of a hat.a. If no one can win more than once, what is the probability that all 3 winners will
Chocolate chips are randomly distributed onto on a bakery's cookies with a rate, on average, of 11 chips per cookie. Assume that the cookies are independent.a. What is the probability that a randomly selected cookie contains exactly 12 chips?b. On average, how many cookies would you have to check
The probability that a student has a donut for breakfast is 0.08. a. If I ask students (at random) from a large university campus whether they had a donut for breakfast, what is the probability the twentieth person I ask will be the first one who had a donut for breakfast? b. How many students do I
Stacey is playing a game of cards using a standard 52-card deck (13 each of hearts, clubs, diamonds, and spades). a. What is the probability the first card she is dealt will be the ace of spades? b. She is dealt 7 cards at the beginning of the game to hold in her hand. What is the probability that
Suppose rain is falling at an average rate of 30 drops per square inch per minute.a. What is the probability that a particular square inch is hit by exactly 4 drops in the next 10-second period?b. How many 10-second intervals do you expect to observe this square inch until you find a 10-second
An exam consists of 20 multiple-choice questions with 5 possible answer choices per question. You haven't studied, so you decide to make random guesses for each answer choice. Assume each question's answer is independent from the others. a. What is the probability that a person who randomly guesses
You are doing laundry, and you are trying to find any small new red socks which may have fallen into the pile of white clothes in the laundry basket by mistake. It's really dark in your basement laundry room, so you have to randomly sample items of clothing. a. In a single laundry basket, there are
You are in charge of coffee bean quality control. In a very large batch of coffee beans, you estimate that the chance a bean is roasted incorrectly is 0.008. You have your employees sample 1000 beans from the seemingly endless supply, to see how many incorrectly roasted beans they find. (The roast
Frederique has 7 dogs: Molly, Ted, Fido, Rocket, Max, Sandy, and Nellie. Assume each dog is independent from the other.a. One of the dogs has stolen her sandwich off her plate at lunch, and she wants to figure out which one. They are each equally likely to have done it, so she is going to randomly
The number of vehicles crossing a line on an interstate is Poisson with an average of 5 per hour. A slug arrives at the interstate and | will wait until no vehicles have crossed the line for 5 minutes until it attempts to get to the other side of the interstate. (This particular slug can count and
There are 5 white mice, 6 brown mice, and 7 gray mice in a cage. For your psychology experiment, you reach in and randomly select 3 to run in a maze. a. What is the expected and standard deviation in the number of white mice which are selected? b. What is the probability that at least 2 of the mice
Fred is a left-handed person who walks into a large lecture room for a 4-hour exam. He's a little absent-minded at the moment because he's worried about the exam. He needs to find a left-handed desk to sit in for the exam or else his back will get cramped up before the exam is over, but he won't
Bethany has a fair six-sided die (with sides 1... 6). Angelica has a fair coin, with "1" on one side and "4" on the other side. Each day, Bethany rolls the die one time, and Angelica flips the coin one time (the results are independent). Let N denote the first day on which Bethany's die has a
An employer has a very large pool of applicants for 8 jobs. The employer needs to report only the sex of the 8 people who are hired (not the names or any other distinguishing features), to a gender-equity review board. How many ways of hiring men and women for these positions are there if:a. The
You fill a mini gumball machine with 60 gumballs: red cherry, pink bubblegum, green lime, and orange, 15 of each. Over time you eat all of the gumballs, two at a time. What is the probability of being left with one pink gumball and one green gumball at the end?
On a certain radio station, 70% of the songs are rock songs, and 30% of the songs are pop songs. The songs are selected independently. Each "block" of songs (a "block" is a set of songs between commercials) contains 10 songs. The DJ says that the next "block" of songs has at least 8 rock songs.
There are 20 ramen noodle packages in a bag. There are 10 beef flavored, and the other 10 are chicken flavored. What is the probability of getting at least one chicken and at least one beef flavor when you grab 3 packages randomly?
A coach is choosing 4 students to go to a badminton competition (the order of selection does not matter). There are 20 and 15 girls to choose from, and the coach chooses randomly so everyone an equal chance of being picked, but nobody can be picked more than once. a. What is the probability that
In a game of chance, you are allowed to shuffle a standard deck of cards and then choose 3 cards randomly. If two or more of them are hearts, then you win.a. What is the probability of winning the game?b. What is the expected number of hearts drawn?
If you randomly assign daughters a, 6, c, d, e to mothers A, B, C, D, E, and you let X be the number of daughters who are correctly assigned to their mother, find the mass of X.
Consider 6 identical balls and 3 distinguishable boxes.a. In how many ways can we put the 6 identical balls into the 3 distinguishable boxes?b. If each box gets at least 1 ball?
A little girl randomly arranges 30 bears-consisting of 10 red bears, 10 yellow bears, and 10 blue bears-into 10 bowls of 3 bears [each. (All outcomes are equally likely.)Let X denotes the number of bowls that have at least one blue bear in the bowl. (The little girl loves blue bears.) Find the
A total of 30 bears-consisting of 10 red bears, 10yellow bears, and 10 blue bears-are randomly arranged in a bucket. 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
Five friends, named Ali, Bob, Charlie, Daniel, and Ebony, each pick a letter from the alphabet on their own (i.e., independently), with all possible outcomes equally likely. For instance, they might pick (starting with Ali's choice) "M,D,P,Z,P."a. What is the probability that they choose 5 distinct
Pierre works 5 days a week. He has 12 shirts, 8 pants, 8 ties, and 4 jackets that he can wear to work. Of these, 4 shirts, 3 nuts, 2 ties, and 2 jackets are blue. Each day he randomly selects one of each item to wear. Assume the selections are independent, and assume his butler launders his clothes
Consider the ways that the letters in the word "Mississippi" can be rearranged (the 4 i's are indistinguishable, the 4 S's are indistinguishable, and the 2 P's are indistinguishable).a. What is the probability that the S's are grouped together?b. What is the probability that the S's are grouped
According to an ad in the newspaper for the local cafe, their Bagel Sandwiches can be "made your way" with "100s of choices for you!" They advertise 7 flavors of bagels, 5 kinds of meat, 3 kinds of cheese, 9 dressings, and 3 veggie selections.a. If you select a sandwich made with one of each:
A pizza place offers the choice of the following toppings: extra cheese, mushrooms, pepperoni, ham, sausage, onions, and green peppers. Assume that each pizza must have at least 1 topping and that order of toppings is irrelevant. Also assume that toppings cannot be reused (double sausage is
A student has 3 math books, 4 history books, 2 chemistry books, and 1 Latin book. He wants to arrange them on a bookshelfa. If all the books have different titles, in how many distinct ways can he arrange them?b. Throughout parts b-e, assume that all the books from a particular topic have the same
Ten people are going to a private screening of a movie in a small theater. There are two rows in the theater. The first row has I seats, and the second row has 6 seats.a. How many ways can you arrange the seating of the 10 people altogether?b. How many seating arrangements are possible if 3 of the
You have a row of 20 books on your desk. You're about to go on vacation and want to grab some good reading material, preferably a good novel or some other work of fiction. Of the 20 books, 12 are fiction and 8 are nonfiction. You grab 5 books at random.a. What is the expected number of fiction
On a "Save The Wetlands" license plate, there are always two letters (e.g., "SW") followed by four digits, and thus 10,000 combinations of plates are available for each pair of letters. If the four digits are selected randomly, and all 10,000 possibilities are equally likely, what is the
In my sock drawer there are 21 white socks, 8 black socks, and 4 brown socks.a. If I randomly pull out 6 socks to take with me on a trip, what is the probability that I pull out one pair of each color?b. What is the probability all the socks are the same color?c. What is the probability that I pull
Rodrigo and Edna are divorced. They are sitting at a circular table with n ≥ 2 chairs altogether, and all seating arrangements are equally likely. What is the probability that they are not sitting next to each other?
If Alice and Alan (a couple) and Barbara and Bob (another couple) sit in a circle of chairs, what is the probability that each of the couples sit together?
If Alice and Alan (a couple) and Barbara and Bob (another couple) and Christine and Charlie (another couple) sit in a circle of chairs, what is the probability that each of the 3 couples sit together?
If Alice and Barbara (two girls) and Alan and Bob (two boys) sit in a circle of chairs, what is the probability that the girls both sit together and both boys sit together?
If Alice, Barbara, and Christine (three girls) and Alan, Bob, and Charlie (three boys) sit in a circle of chairs, what is the probability that the girls all sit together and the boys all sit together?
If Alice and Barbara (two girls) and Alan and Bob (two boys) sit in a circle of chairs, what is the probability that none of the girls are adjacent and none of the boys are adjacent?
If Alice, Barbara, and Christine (three girls) and Alan, Bob, and Charlie (three boys) sit in a circle of chairs, what is the probability that none of the girls are adjacent and none of the boys are adjacent?
If n girls and n boys sit in a circle of chairs, what is the probability that none of the girls are adjacent and none of the boys are adjacent?
A professor buys postage stamps and puts them in an envelope. Exactly 30 of the stamps are for letters, and exactly 10 of the stamps are for postcards. The professor's wife grabs two stamps. All outcomes are equally likely.a. What is the probability that both stamps she selects are for letters?b.
Identify from the information below whether X is a discrete or continuous random variable. a. Let X be the height of a randomly selected 8-year-old child. b. Let X be a random variable that takes values on the nonnegative integers, i.e X ( {0,1,2,3,...}. c. Let X be a random variable that takes
Consider the function from Example 24.13, i.e., fx(x) = 3e-3x for x > 0, and fx (x) = 0 otherwise.a. What is the cumulative distribution function Fx(x)?b. Draw a picture of the cumulative distribution function Fx(x).
LetfY(y) = l2y2(l-y),for 0 ≤ y ≤ l,and fY(y) = 0 otherwise. Find the CDF of Y.
Suppose that X has CDFFind the density fx(x).
LetFY(y) = √y,For 0 ≤ y ≤ 1and FY(y) = 0 for y < 0, and FY(y) = 1 for y > 1. Find the density of Y.
Suppose a density fx (x) increases linearly from (16,0) to (24,1/4).a. Find the CDF of X.b. What is the value of a so that P(X > a) = 0.75?
Identify from the information below whether X is a discrete or continuous random variable. a. Let X be a random variable that takes values on the set {3,4,5,6,7} b. Let X be the number of defective light bulbs in a shipment. c. Let X be a random variable that takes values on the real interval [3,7]
What is the constant k that makes the following function a valid density?
What is the constant k that makes the following function a valid density?
Suppose that X has CDFa. Find the density fx (x).b. Find the median of X.
Again consider the function from Example 24.13, i.e., let fx (x) = 3e-3x for x > 0, and fx (x) = 0 otherwise. Evaluate
Let X be the lifetime (in years) of a carbon-14 atom before it decays. Then X has densityfx (x) = ln2/5730 exp ( - ln2/5730 x). for 0 < x.And fx (x) = 0 otherwise. Find the length of time a (in years) so thatP(X ≤ a) = l/2 and P(X > a) = 1/2.
Let X have densityfx (x) = 1/2 x2e-x, for x > 0,and fx (x) = 0 otherwise. Find P(X < 2)
Is it possible to define two random variables X and Y in such a way that the CDF for X is always strictly larger than the CDF for V? In other words, is it possible to have Fx(x) > FY(y) always? If yes, then give an example. If not, then explain why not.
Let X be a Continuous Uniform random variable on the interval [a, b]. Compute E (Xn).
If you know P(X > 2) = 0.3, fill in the chart for the other values you know. Write "???" if there is not enough information to figure out a value.
If you know P(X
Let X have density fx(x) = kx2 (l - x)2 for 0 ≤ x ≤ 1, and fx(x) = 0 otherwise, where k is constant.a. Find the value of k.b. Find P(X > 3/4).
Let X have densityfx (x) = l/5,for 2 ≤ x ≤ 7and fx (x) = 0 otherwise.a. Find P{X > 3).b. Find the CDF Fx (x).c. Find P(X < 5).d. Find P(X > 8).e. Find P(5 ≤ X ≤ 6.5).f. Find P(X < 1).g. Graph the density fx(x).h. Graph the CDF Fx (x).
Let X be the waiting time (in minutes) until a student's friend arrives. Suppose that X has density fx(x) = 1/3 e-x/3 for 0 < x, and fx{x) = 0 otherwise. a. Find P(3 ≤ X v 6). b. Find the CDF Fx(x). c. Find P{X ≥ 24). d. Find P(X ≤ -3). e. Find P(3 ≤ X ≤ 12). f. Find P(X < 300). g. Graph
Consider the function from Example 24.12, i.e., fx(x) = 1/6 for 4 < x < 10, and fx(x) = 0 otherwise.a. What is the cumulative distribution function Fx(x)?b. Draw a picture of the cumulative distribution function Fx (x).
Consider a pair of random variables X, Y with constant joint density on the triangle with vertices at (0, 0), (3, 0), and (0, 3). Find P (X + Y > 2).
Consider X, Y with joint densityfX,Y(x, y) = 1/2500(10 - x)(10 -y),for 0 ≤ x ≤ 10, 0 ≤ y ≤ 10,and fxy(x,y) = 0 otherwise.a. Find P(X < 9,y < 9).b. Find P(X < 9,y > 9).c. Find P(X > 9,Y > 9).d. Check that the answers from parts a-d sum to 1. Exercise 11. Let X, Y have joint
Let X, Y have joint density fX, Y(x, y) = x + y, for 0 ≤ x,y ≤ 1, and fx, y (x, y) = 0 otherwise. Find the joint CDF, i.e., Fx, y (x, y) = P(X ≤ x, Y≤ y).

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