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

I am at a dog training class. Twenty percent of the dogs are hounds. The other eighty percent of the dogs are of the toy variety. Ninety percent of the hounds are stubborn and difficult to train. Thirty percent of the dogs in general are stubborn. What is the probability that a stubborn dog is a
Eighty percent of all mp3 players are iPods. Five percent of iPods are defective. Seven percent of all mp3 players are defective. A randomly chosen mp3 player is taken to a repair shop because it is defective. What is the probability it is an iPod?
In the Department of Mathematical and Computational Science, 83% of lecture halls have chalk¬boards, and the other 17% have dry-erase boards. Of the classes taught in rooms with chalkboards, 75% are mathematics courses, 15% are computer sci¬ence courses, and 10% are statistics courses. Of the
In the theory of general intelligence, it is stated that being good in one intelligence, like math, increases the chance of one being good in another intelligence. Suppose 10% of the people are good at art, and that 40% of the people who are good at art are also good at math. If a person is not
A student is chosen at random. You want to know the probability that the student has an iPod. This could be hard to determine since there are over 40,000 students on the given campus. Fortunately, a current survey recorded that 47% of first-year students have iPods, and you know that 32% of the
It is estimated that 82% of homes have working smoke detectors. On average, 22% of fires result in fatalities, but the presence of a working smoke detector cuts the risk to just 7%.a. If a random fire resulted in a fatality, what is the probability that the house had a working smoke detector?b. In
Consider the following game: The player flips a fair coin. If it shows a head, he gets to roll a 4-sided die. If it shows a tail, he gets to roll a 6-sided die. In either case, let A denote the event that he gets 1 on the die roll. Let H denote the event that his coin flip shows a head, i.e., that
The probability of A is 0.03, and the probability of C is 0.27. With the given information, what are potential probabilities of B and D?
A customer waits for a bus to appear.a. List 5 possible outcomes.b. What is the sample space?c. Write a partition for the customer's waiting time, using five-minute in-tervals for the partition. (Assume that the customer can wait as long as needed for the bus.)d. Explain how the answer to part c
Randomly open a 300-page book, and mark a page. What is the probability that the number of the page contains the lucky number 5?
There are 45 egg boxes in a store. Twenty are Brand A, fifteen are brand B, and ten are Brand C. Brand A and C each have half green boxes and half yellow boxes. Brand B is all yellow. a. If you have Brand B or C, what is the chance that you have a green box? b. If you have a yellow box, what is the
According to the National Insti-tute of Mental Health (from in any year, 18.1% of all U.S. adults suffer from anxiety disorders. Among people who have anxiety disorders, 30.2% are between 18 and 29 years old. According to the 2010 U.S. Census, approximately 10% of the U.S. adults are between 18 and
Consider a collection of 100 light bulbs. Let Xj denote the lifetime of the jth bulb. Let Y denote the sum of the lifetimes of all 100 bulbs.
A pile of puzzle pieces falls onto the floor. Let X denotes the number of pieces that are edge pieces, and let Y denote the number of pieces that are interior (i.e., not edge) pieces.
A half-gallon container of milk is tested by a quality control company. Let X be a rating of the milk quality, given as an integer from 1 to 10; let Y denote the exact volume of milk in the selected half-gallon container; let Z be "1" if the half-gallon of milk is 1% milkfat, or "2" if it is 2%
Roll a pair of dice until "snake eyes" (i.e., a pail of l's) appear. Let X denote the total number of rolls required. Let Y denote the sum of all of the dice rolled during this process.
Select a random student from your course; let X denote the age, in days, of the selected student; let Y denote the length of the student's name.
When opening a container of 32 toy bal¬let dancers performing pirouettes, let X denote the number of broken dancers, and let Y denote the number of whole (unbroken) dancers.Write Z = X/Y, i.e., Z is the ratio of broken dancers to whole dancers. What complication arises in this definition of Z as a
Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously, i.e., grab two cards at once-so they are not the same card!). There are 12 cards which are considered face cards (4 Jacks, 4 Queens, 4 Kings). Let X be the number of face cards that you get. a. Find P{X =
Pick a basket of onions. Let X be the weight of the onions (in pounds). Let Y be the number of onions in the basket.
Let X be the cumulative grade point average of a randomly chosen student. Let Y be the score on the student's most recent exam. Let Z be the number of exams that the student takes during the current semester.
When ordering a new box of harmonicas, let X denote the time (in hours) until the box arrives, and let Y denote the number of harmonicas that work properly.
Select a random student's music player. Let X denote the number of blues songs on the music player; let Y denote the number of jazz songs; let Z denote the number of rock songs.
Consider the time X (in min¬utes) from now until your boyfriend/girlfriend calls on the telephone. Let Y be the length of the call. Let Z be the number of times that he/she calls in one evening.
Consider a box of two light bulbs. Let X be the lifetime of the first bulb removed from the box, and let Y denote the lifetime of the second bulb removed from the box.
A student makes a trip once per day to the store, and he always buys a snack. The student eats the snack 70% of the time, but the other 30% of the time, his roommate eats it first. a. During a period of four days, the student keeps track of whether he gets to eat his snack. What is the sample space
Consider a random variable X that has CDFWhat is the mass of X?
Consider a random variable X that has CDFWhat is the mass of X?
Alice, Bob, and Charlotte are looking for butter-flies. They look in three separate parts of a field, so that their probabilities of success are independent.• Alice finds 1 butterfly with probability 17%, and otherwise does not find one.• Bob finds 1 butterfly with probability 25%, and
At a restaurant that sells appetizers: • 8% of the appetizers cost $1 each, • 20% of the appetizers cost $2 each, • 32% of the appetizers cost $3 each, • 40% of the appetizers cost $4 each. An appetizer is chosen at random, and X is its price. Draw the CDF of X.
Roll two 4-sided dice and let X denote the sum. a. Draw the mass of X. b. Draw the CDF of X.
Suppose X has massand px(x) = 0 otherwise.a. For A = 2, make a plot of the probability mass function.b. What is the CDF of X, when A = 2?c. Make a plot of the CDF, when A = 2.For future reference, this is called a Poisson random variable. You may want to refer to the Math Review for help with the
Flip a coin three times. Let X denote the number of heads minus the number of tails. So, for instance, if (if, T, T) is the outcome, then X = 1- 2 = -1.a. What are the possible values of X?b. What is the mass of X?c. Make a plot of the probability mass function.d. What is the CDF of X?e. Make a
Using a shuffled standard deck of 52 playing cards, a magician wants to do a trick where he tries to guess which card an audience member has selected if the audience member chooses a card at random and doesn't show the magician. Let X be the number of cards the magician will guess correctly if he
Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is 1/3. Let X be the number of attempts required. If he is not successful within tin first 5 attempts, then
Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously, i.e., grab two (aids at once-so they are not the same card!). There are 12 cards which are considered face cards (4 Jacks, 4 Queens, 4 Kings). Let X be the number of face cards that you get. Draw the CDF Fx(x)
As in Exercise 2.21, a sequence of seven people walk into a post office and only their sexes are noted. Assume that each of the seven customers is equally likely to be a man or a woman. Let X denote the number of customers that are female.a. Find the mass of X.b. Find the CDF of X.
As in Exercises 2.1 and 4.3, and in Ex-ample 3.11, a randomly chosen song is from the blues genre with probability 330/27333; from the jazz genre with probability 537/27333; from the rock genre with probability 8286/27333; or from some other genre with probability 18180/27333. Let X be "1" if a
Suppose that 13% of all milk cartons have 1% milkfat, and 28% of all milk cartons have 2% milkfat, and 18% of all milk cartons are fat-free, and 41% of all milk cartons are of some other type. Randomly choose a carton of milk, and let X be "1" if the carton of milk is 1% milkfat, or "2" if the
The Super Breakfast Challenge (SBC) consists of bacon, eggs, oatmeal, orange juice, milk, and several other foods, and it costs $12.99 per person to order at a local restaurant. It is known to be very difficult to consume the entire SBC. Only 10% of people are able to eat all of the SBC. The other
A cereal company puts a Star Wars toy watch in each of its boxes as a sales promotion. Twenty percent of the cereal boxes contain a watch with Obi Wan Kenobi on it. You are a huge Obi Wan fan, so you decide to buy 8 boxes of the cereal in hopes that you will find an Obi Wan watch. Let X denote the
Recall that a standard deck of 52 playing cards has 4 suits (hearts, spades, clubs, and diamonds) and 13 cards in each suit (labeled 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A). The cards are shuffled. You are dealt the first 5 cards off the top of the deck. Let X be the number of hearts you get.a.
Suppose X is a discrete random variable with a probability mass function px (x) - c(4 - x) for x in { - 1,2,3}, and Px (x) = 0 otherwise.a. What is the value of c so that px (x) is a mass?b. Make a plot of the probability mass function.c. What is the CDF of X?d. Make a plot of the CDF.
Suppose X is a discrete random variable with masspx(x) = l/4 for x = 1,2,3,4,and px(x) = 0 otherwise. For future reference, this is a called a Discrete Uniform distribution. We will study these random variables more in Chapter 20.a. Make a plot of the probability mass function.b. What is the CDF of
Ten students apply for a job opening, but only 1 of the students will be selected. The employer choc-' randomly; all ten outcomes are equally likely. If person 3, 5, 7, or 9 gets the job, let X = 1; otherwise, X = 0. If person 1, 2, 3, 4, or 5 gets the job, lei Y = 1; otherwise, Y = 0. Are X and Y
Consider some 4-sided dice. Roll two of these dice. Let X denote the minimum of the two values that appear, and let Y denote the maximum of the two values that appear. a. Find the joint mass pX,Y(x,y) of X and Y. b. Find the joint CDF FX,Y(x,y) of X and Y. It suffices to give the values FX,Y(x, y)
Prove that the statements of independence in Definition 9.13 are equivalent.
Each day, Maude has a 1% chance losing her cell phone (her behavior on different days is independent). Each day, Maude has a 3% chance of forgetting to eat breakfast (again, her behavior on different days is independent). Her breakfast and cell phone habits an independent. Let X be the number of
Alice, Bob, and Charlotte are looking for butter-flies. They look in three separate parts of a field, so that their probabilities of success do not affect each other. • Alice finds 1 butterfly with probability 17%, and otherwise does not find one. • Bob finds 1 butterfly with probability 25%,
Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously, i.e., grab two cards at once-so they are not the same card!). There are 12 cards which are considered face cards (4 Jacks, 4 Queens, 4 Kings). There are 4 cards with the value 10. Let X be the number of face
Roll two dice, one colored red and one colored blue. Let Y denote the maximum value that appears on the two dice. Let X denote the value of the blue die. Find the conditional mass of X given Y.
Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is 1/3. Let X be the number of attempts required. If he is not successful within the first 5 attempts, then
As in Exercise 3.1, Jack and Jill are indepen-dently struggling to pass their last (one) class required for graduation. Jack needs to pass Calculus III, but he only has probability 0.30 of passing. Jill needs to pass Advanced Pharmaceuticals, but she only has probability 0.46 of passing. They work
Alice, Bob, and Charlotte are looking for but-terflies. They look in three separate parts of a field, so that their probabilities of success do not affect each other.• Alice finds 1 butterfly with probability 17%, and otherwise does not find one.• Bob finds 1 butterfly with probability 25%, and
A student rolls a die until the first "4" appears. Let X be the numbers of rolls required until (and including) this first "4." After this is completed, he begins rolling again until he gets a "3." Let Y be the number of rolls, after the first "4," up to (and including) the next "3." E.g., if the
Consider some 4-sided dice. Roll two of these dice. Let X denote the minimum of the two values that appear, and let Y denote the maximum of the two values that appear. a. Find the expected value of X. b. Find the expected value of Y.
Pick two cards. Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously, i.e., grab two cards at once-so they are not the same card!). There are 12 cards which are considered face cards (4 Jacks, 4 Queens, 4 Kings). There are 4 cards with the value 10. Let X be the
Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is 1/3. Let X be the number of attempts required. If he is not successful within the first 5 attempts, then
Jorge has three kids who spotted a Claw machine with toys they want. In stores the toys costs $10 each, but each play on the claw only costs $1. The probability of Jorge winning a game on the Claw is 0.12. Should he use the Claw to get the toys (he needs one toy per kid), or does he expect it to be
Two fair dice are rolled. Let X be the sum of the dice. What is the expected value of X?
As in Exercise 3.2, Four students order noodles at a certain local restaurant. Their orders are placed independently. Each student is known to prefer Japanese pan noodles 40% of the time. How many of them do we expect to order Japanese pan noodles?
In a bowling alley, 20% of the time when someone bowls, he or she gets a strike. If there are 3 people in the bowling alley, and X is the total number of people who get a strike on their current attempt, what is the expected value of X?
There is a bowl containing 30 cashews, 20 pecans, "5 almonds, and 25 walnuts. I am going to randomly pick and eat 3 nuts. What is the expected number of cashews I will eat?
Let X be the number of games of chess I win against Stephen. Assume that I have a 30% chance of winning in any particular game against him, and we play 5 games, with outcomes assumed to be independent. Find the expected value of X.
As in Exercise 10.1, Jack and Jill are independently struggling to pass their last (one) class required for graduation. Jack needs to pass Calculus III, but he only has probability 0.30 of passing. Jill needs to pass Advanced Pharmaceuticals, but she only has probability 0.46 of passing. They work
Alice, Bob, and Charlotte are looking for but-terflies. They look in three separate parts of a field, so that their probabilities of success do not affect each other.• Alice finds 1 butterfly with probability 17%, and otherwise does not find one.• Bob finds 1 butterfly with probability 25%, and
A student rolls a die until the first "4" appears. Let X be the numbers of rolls required until (and including) this first "4." After this is completed, he begins rolling again until he gets a "3." Let Y be the number of rolls, after the first "4," up to (and including) the next "3." E.g., if the
Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is 1/3. Let X be the number of attempts required. If he is not successful within the first 5 attempts, then
Consider some 4-sided dice. Roll two of these dice. Let X denote the minimum of the two values that appear, and let Y denote the maximum of the two values that appear. Let Aj be the event containing all outcomes in which the minimum of the two dice is "j or greater." Let Xj indicate whether Aj
Pick two cards. Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously, i.e., grab two cards at (once-so they are not the same card!). There are 12 cards which are considered !face cards (4 Jacks, 4 Queens, 4 Kings). There are 4 cards with the value 10. Let X be
Flip a coin until the second head comes up. Let X be the number of flips needed to get the second head. What is the E(X)?
Michael puts his iTunes on shuffle mode where songs are not allowed to be replayed. He has 2,781 songs saved in iTunes, and exactly one of these is his favorite. a. How many songs is he expected to have to listen to until his very favorite song comes up? b. Now suppose that he allows songs to be
A little girl has a 96-pack of crayons. She picks up crayons, at random, to check the color, and she leaves them in a separate pile after inspecting the color. She pulls crayons out of the pack until she gets the sea foam green crayon. a. What is the expected number of crayons she will check until
Consider a group of 120 students. They are split into two lectures (60 students each). They are also split into six labs (20 students each). All assignments of students to lectures and labs are equally likely. How many classmates does Barry expect to be in both his lecture and his lab too?
As in Exercise 10.2, four students order noodles at a certain local restaurant. Their orders are placed independently. Each student is known to prefer Japanese pan noodles 40% of the time. How many of them do we expect to order Japanese pan noodles? Let A1, A2, A3, A4 be the events that
Eighty-seven percent of people tear up when cutting an onion. People are selected randomly and independently to cut up an onion. What is the expected number of people who cut up an onion until you find the third person who does it without crying?
Eight people enter an elevator in the parking garage below a building. Each person chooses her exit independently of the other people. The building has floors 1 through 10. What is the expected number of stops that the elevator makes?
A total of 30 bears-consisting of 10 red bears, 10 yellow bears, and 10 blue bears-are randomly arranged into 10 groups of 3 bears each. Compute the expected number of groups in which all 3 bears are different colors.
Consider n pieces of rope. Each piece is colored blue at one end and red at the other. The blue ends of the ropes are randomly paired with the red ends of the ropes and tied together, one-to-one, i.e., all n! Such possible methods of joining the ropes this way are equally likely. Let X be the
Three hundred little plastic yellow ducks are dumped in a pond; one of them contains a prize stamped on the bottom. Leonardo examines each duck until he discovers the prize. He discards each duck without a prize after he checks it, so that he never needs to check a duck more than one time. How many
A weather forecasting program gets the daily pre-dictions right about 87% of the time. Assuming each day is independent, what is the expected number of days that will pass until the program gets the forecast wrong?
A man sits down to watch a movie. 40% of his options are action films, 35% are comedies, and 25% are drama. He wants to watch a comedy and starts picking movies at random from his box. If previous picks are put back in the box after marking the choice, how many movies is he expected to have to pick
Two three-partitioned spinners are spun. Each of the three parts (labeled 1, 2, and 3) have an equally likelihood of occurring. Let X be the maximum of two spins. a. Find E(X). b. Find(X2). c. Find Vary(X).
You purchase 8 raffle tickets at the county fair. Each ticket costs $5. A ticket is worth $100 with probability 1/400, but is worthless otherwise. What is the expected value of your purchase? Be sure to take into account the original purchase price of the tickets.
Suppose that, among a certain group of students, SAT scores have a mean value of 1026 and a standard deviation of 209. Let X denote a randomly chosen student's score. What is E(X2)?
Alice, Bob, and Charlotte are looking for butterflies. They look in three separate parts of a field, so that their probabilities of success do not affect each other. • Alice finds 1 butterfly with probability 17%, and otherwise does not find one. • Bob finds 1 butterfly with probability 25%,
At a restaurant that sells appetizers: • 8% of the appetizers cost $1 each, • 20% of the appetizers cost $2 each, • 32% of the appetizers cost $3 each, • 40% of the appetizers cost $4 each. An appetizer is chosen at random, and X is its price. Each appetizer has 7% sales tax. So Y - 1.07X
In each round of a game, you earn a dollar if a die show 1 or 2, you lose a dollar if a die shows 5 or 6, and you neither earn nor lose anything if a die shows 3 or 4. Let Xj be +1, -1, or 0 according to your outcome on the jth round. Let X = X1 + X2 + . . . + X10. Find the variance of X.
A student gets at least 8 hours of sleep 45% of the nights; the sleeping schedule is independent from night to night. Let X1, X2, X3, X4 indicate whether the student gets at least 8 hours of sleep during the next four nights respectively. Let X = X1 + X2 + X3 + X4. Find the variance of X.
A magician wants to do a trick where he tries to guess the value of the card that an audience member has selected (the suit of the card is not taken into account). He has six decks of cards so he gives one deck to each of six audience members. Let X denote the number of cards that he guesses
At a concert on campus, 20% of people purchase Zone A tickets, for $47 each. The other 80% of people purchase Zone B tickets, for $38 each. If five people are selected at random, what is the variance of the revenue from these five ticket sales?
As in Example 12.16, Jim and his brother both like chocolate chip cookies best. They have a jar of cookies with 5 chocolate chip cookies, 3 oatmeal cookies, and 4 peanut butter cookies. They are each allowed to have 3 cookies. To be fair, they agree to randomly select their cookies without peeking,
Consider a random variable X with massa. Find the expected value of 1/X.b. Find the standard deviation of 1/X.
In a dice game, a "Yahtzee" is a result in which all five dice within a round have the same value. To simplify this problem, assume that the five dice are just rolled one time per round. (In the actual Yahtzee game, dice can be re-rolled.) Let X be the number of times that a player gets a Yahtzee
Roll a die and flip a coin. Let Y be the value of the die. Let Z = 1 if the coin shows a head, and Z = 0 otherwise. Let X = Y + Z. Find the variance of X.
A student was at work at the county amphitheater, and was given the task of cleaning 1500 seats. To make the job more interesting, his boss hid a golden ticket somewhere in the seats. The ticket is equally likely to be in any of the seats. Let X be the number of seats cleaned until the ticket is
Two dice are rolled. Let X be the value of the first die minus the second. Find Var(X).
Consider the massa. Find the average value of X.b. Find the average value of 4X.c. Find the average value of X4.d. Find the average value of Ax.e. Find the variance of X.f. Find the variance of 4X.g. Find the variance of X4.h. Find the variance of 4X.
Consider the massa. Find E{X).b. Find E(2/X).c. Find 2/E(X).d. Find E(|X|).e. Find VarpQ.f. Find Var(|X|).
If X and Y have joint density fX,Y(x, y) - 8xy on the triangle 0 < y < x < 1, find E(XY).

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