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elementary probability for applications

Elementary Probability For Applications 1st Edition Rick Durrett - Solutions

A car rental company has rental offices at both Kennedy and LaGuardia airports. Assume that a car rented at one airport must be returned to one of the two airports. If the car was rented at LaGuardia the probability that it will be returned there is 0.8; for Kennedy the probability is 0.7. Suppose
The 1990 census showed that 36% of the households in the District of Columbia were homeowners, while the remainder were renters. During the next decade 6% of the homeowners became renters and 12% of the renters became homeowners. (a) What percentage were homeowners in 2000? In 2010? (b) If these
Most railroad cars are owned by individual railroad companies. When a car leaves its home railroad’s trackage, it becomes part of the national pool of cars and can be used by other railroads. A particular railroad found that each month 15% of its boxcars on its home trackage left to join the
A rapid transit system has just started operating. In the first month of operation, it was found that 25% of commuters are using the system, while 75% are traveling by automobile. Suppose that each month 10% of transit users go back to using their cars, while 30% of automobile users switch to the
A regional health study indicates that from one year to the next, 75% of smokers will continue to smoke, while 25% will quit. 8% of those who stopped smoking will resume smoking, while 92% will not. If 70% of the population were smokers in 1995, what fraction will be smokers in 1998? In 2005? In
The town of Mythica has a “free bikes for the people program.” You can pick up bikes at the library (L), the coffee shop (C), or the cooperative grocery store (G). The director of the program has determined that if a bike is picked up at the library ends up at the coffee shop with probability
Bob eats lunch at the campus food court every week day. He either eats Chinese food, Quesadilla, or Salad. His transition matrix isHe had Chinese food on Monday. What are the probabilities for his three meal choices on Friday (4 days later). C 0.15 0.6 0.1 0.3 QS C Q 0.4 S 0.1 0.25 0.5 0.6
Census results reveal that in the United States 80% of the daughters of working women work and that 30% of the daughters of nonworking women work. (a) Write the transition probability for this model. (b) In the long run what fraction of women will be working?
Three of every four trucks on the road are followed by a car, while only one of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?
In a test paper the questions are arranged so that 3/4’s of the time a true answer is followed by a true, while 2/3’s of the time a false answer is followed by a false. You are confronted with a 100-question test paper. Approximately what fraction of the answers will be true?
When a basketball player makes a shot then he tries a harder shot the next time and hits (H)with probability 0.4 and misses (M)with probability 0.6.When he misses he is more conservative the next time and hits (H) with probability 0.7 and misses (M) with probability 0.3.(a) Write the transition
A regional health study shows that from1 year to the next 76% of the people who smoked will continue to smoke and 24% will quite. 8% of those who do not smoke will start smoking, while 92% of those who do not smoke will continue to be nonsmokers. In the long run what fraction of people will be
In unprofitable times corporations sometimes suspend dividend payments. Suppose that after a dividend has been paid the next one will be paid with probability 0.9,while after a dividend is suspended the next one will be suspended with probability 0.6. In the long run what is the fraction of
A university computer room has 30 terminals. Each day there is a 3% chance that a given terminal will break and a 72% chance that a given broken terminal will be repaired. Assume that the fates of the various terminals are independent. In the long run what is the distribution of the number of
A plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each of these genotypes is crossed with a pink (RW) plant then the offspring fractions areWhat is the long-run fraction of plants of the three types? RR RR 0.5 RW 0.25 WW 0 RW 0.5 0.5 0.5 ww
A certain town never has two sunny days in a row. Each day is classified as rainy, cloudy, or sunny. If it is sunny one day then it is equally likely to be cloudy or rainy the next. If it is cloudy or rainy then it remains the same 1/2 of the time, but if it changes it will go to either of the
Amidwestern university has three types of health plans: a health maintenance organization (HMO), a preferred provider organization (PPO), and a traditional fee for service plan (FFS). In 2000, the percentages for the three plans were HMO: 30%, PPO: 25%, and FFS: 45%. Experience dictates that people
A sociologist studying living patterns in a certain region determines that the pattern of movement between urban (U), suburban (S), and rural areas (R) is given by the following transition matrix:In the long run what fraction of the population will live in the three areas? US U 0.86 S 0.05 R 0.03 R
In a large metropolitan area, commuters either drive alone (A), carpool (C), or take public transportation (T). A study showed that 80% of those who drive alone will continue to do so next year, while 15%will switch to carpooling and 5% will use public transportation. 90% of those who carpool will
In a particular county voters declare themselves as members of the Republican, Democrat, or Green party. No voters change directly from the Republican to Green party or vice versa. In a given year 15% of Republicans and 5%ofGreen party members will become Democrats, while 5% of Democrats switch to
(a) Three telephone companies A, B, and C compete for customers. Each year A loses 5% of its customers to B and 20% to C; B loses 15% of its customers to A and 20% to C; C loses 5% of its customers to A and 10% to B.(a) Write the transition matrix for the model.(b) What is the limiting market
An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. No one moves from poor to preferred or from preferred to poor in 1 year. 40% of the customers in the poor category become satisfactory, 30% of those in the satisfactory category move to
A professor has two lightbulbs in his garage. When both are burned out, they are replaced, and the next day starts with two working lightbulbs. Suppose that when both are working, one of the two will go out with probability 0.02. (Each has probability 0.01 and we ignore the possibility of losing
An individual has three umbrellas; some at her office and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will take an umbrella, if one is there. Otherwise, she gets wet. Assume that independent of the past, it rains on each trip with
At the end of a month, a large retail store classifies each of its customer’s accounts according to current (0), 30–60 days overdue (1), 60–90 days overdue (2), more than 90 days overdue (3). Their experience indicates that the accounts move from state to state according to a Markov chain
At the beginning of each day, a piece of equipment is inspected to determine its working condition, which is classified as state 1 = new, 2, 3, or 4 = broken. We assume the state is a Markov chain with the following transition matrix:(a) Suppose that a broken machine requires 3 days to fix. To
To make a crude model of a forest we might introduce states 0 = grass, 1 = bushes, 2 = small trees, 3 = large trees, and write down a transition matrix such as the following:The idea behind this matrix is that if left undisturbed a grassy area will see bushes grow, and then small trees, which of
Five white balls and five black balls are distributed in two urns in such a way that each urn contains five balls. At each step we draw one ball from each urn and exchange them. Let Xn be the number of white balls in the left urn at time n. (a) Compute the transition probability for Xn. (b) Find
Two competing companies are trying to buy up all the farms in a certain area to build houses. In each year 10% of farmers sell to company 1, 20% sell to company 2, and 70% keep farming. Neither company ever sells any of the farms that they own. Eventually all the farms will be sold. How many will
A warehouse has a capacity to hold four items. If the warehouse is neither full nor empty, the number of items in the warehouse changes whenever a new item is produced or an item is sold. Suppose that (no matter when we look) the probability that the next event is “a new item is produced” is
The Macro soft Company gives each of its employees the title of programmer (P) or project manager (M). In any given year 70% of programmers remain in that position, 20% are promoted to project manager, and 10% are fired (state X). 95% of project managers remain in that position, while 5% are fired.
At a nationwide travel agency, newly hired employees are classified as beginners (B). Every 6 months the performance of each agent is reviewed. Past records indicate that transitions through the ranks to intermediate (I) and qualified (Q) are according to the following Markov chain, where F
At a manufacturing plant, employees are classified as trainee (R), technician (T), or supervisor (S). Writing Q for an employee who quits we model their progress through the ranks as a Markov chain with transition probability(a) What fraction of recruits eventually make supervisor?(b) What is the
The two previous problems have the following form:Show that(a) The probability of being absorbed in A rather than B is ac/(a + b)(c + d) and(b) The expected time to absorption starting from 1 is 1/(a + b) + a/(a + b)(c + d). 1 1 l - a - b 2 0 A 0 B 0 2 a 1 - c - d 0 0 A 0 C 1 0 B b d 0 1
Suppose three runners from team A and three runners from team B have a race. If all six runners have equal ability, what is the probability that the three runners from team A will finish first, second, and fourth?
Four men and four women are shipwrecked on a tropical island. How many ways can they(a) Form four male–female couples,(b) Get married if we keep track of the order in which the weddings occur,(c) Divide themselves into four unnumbered pairs,(d) Split up into four groups of two to search the
Two balls are drawn from an urn with balls numbered from1 up to 10.What is the probability that the two numbers will differ by more (>) than 3?
How can 5 black and 5 white balls be put into two urns to maximize the probability that a white ball is drawn when we draw from a randomly chosen urn?
Suppose we draw k cards out of a deck. What is the probability that we do not draw an ace? Is the answer larger or smaller than (3/4)k?
You and a friend each roll two dice. What is the probability that you both have the same two numbers?
In a dice game the “dealer” rolls two dice, the player rolls two dice, and the player wins if his total is larger than the dealer’s. What is the probability that the player wins?
What is the most likely total for the sum of four dice and what is its probability?
Charlie draws 5 cards out of a deck of 52. If he gets at least three of one suit, he discards the cards not of that suit and then draws until he again has 5 cards. For example, if he gets 3 hearts, 1 club, and 1 spade, he throws the 2 non hearts away and draws 2 more. What is the probability that
Suppose 60% of the people in a town will get exposed to flu in the next month. If you are exposed and not inoculated then the probability of your getting the flu is 80%, but if you are inoculated that probability drops to 15%. Of two executives at Beta Company, one is inoculated and one is not.
John takes the bus with probability 0.3 and the subway with probability 0.7. He is late 40% of the time when he takes the bus, but only 20% of the time when he takes the subway. What is the probability that he is late for work?
The population of Cyprus is 70% Greek and 30% Turkish. 20% of the Greeks and 10% of the Turks speak English. What fraction of the people of Cyprus speak English?
You are going to meet a friend at the airport. Your experience tells you that the plane is late 70% of the time when it rains, but is late only 20% of the time when it does not rain. The weather forecast that morning calls for a 40% chance of rain. What is the probability that the plane will be
Two boys have identical piggy banks. The older boy has 18 quarters and 12 dimes in his; the younger boy, 2 quarters and 8 dimes. One day the two banks get mixed up. You pick up a bank at random and shake it until a coin comes out.What is the probability that you get a quarter? Note that there are
Suppose that the number of children in a family has the following distribution:Assume that each child is independently a girl or a boy with probability 1/2 each.If a family is picked at random what is the chance it has exactly two girls. Number of children Probability 0 1 2 0.15 0.25 3 4 0.3 0.2 0.1
A student is taking a multiple-choice test in which each question has four possible answers. She knows the answers to 50% of the questions, can narrow the choices down to two 30% of the time, and does not know anything about 20% of the questions. What is the probability that she will correctly
Three boys take turns shooting a basketball and have probabilities 0.2, 0.3,and 0.5 of scoring a basket. Compute the probabilities for each boy to get the first basket.
Change the second and third probabilities in the last problem so that each boy has an equal chance of winning.
The alpha fetal protein test is meant to detect spina bifida in unborn babies, a condition that affects 1 out of 1,000 children who are born. The literature on the test indicates that 5% of the time a healthy baby will cause a positive reaction. We will assume that the test is positive 100% of the
Binary digits, that is, 0’s and 1’s, are sent down a noisy communications channel. They are received as sent with probability 0.9 but errors occur with probability 0.1.Assuming that 0’s and 1’s are equally likely,what is the probability that a 1 was sent given that we received a 1?
Two hunters shoot at a deer, which is hit by exactly one bullet. If the first hunter hits his target with probability 0.3 and the second with probability 0.6, what is the probability that the second hunter killed the deer? The answer is not 2/3. Do you think the answer is larger or smaller?
A cab was involved in a hit-and-run accident at night. Two cab companies green and blue operate 85% and 15% of the cabs in the city respectively. A witness identified the cab as blue. However, in a test only 80% of witnesses were able to correctly identify the cab color. Given this what is the
A student goes to class on a snowy day with probability 0.4, but on a non snowy day attends with probability 0.7. Suppose that 20% of the days in February are snowy. What is the probability that it snowed on February 7 given that the student was in class on that day?
A company gave a test to 100 salesman, 80 with good sales records and 20 with poor sales records. 60% of the good salesman passed the test but only 30% of the poor salesmen did. Andy passed the test. Given this, what is the probability that he is a good salesman?
A company rates 80% of its employees as satisfactory and 20% as unsatisfactory. Personnel records indicate that 70% of the satisfactory workers had prior experience but only 40% of the unsatisfactory workers did. If a person with previous work experience is hired, what is the probability that they
A golfer hits his drive in the fairway with probability 0.7. When he hits his drive in the fairway he makes par 80% of the time. When he doesn’t, he makes par only 30% of the time. He just made par on a hole. What is the probability that he hit his drive in the fairway?
You are about to have an interview for Harvard Law School. 60% of the interviewers are conservative and 40% are liberal. 50% of the conservatives smoke cigars but only 25% of the liberals do. Your interviewer lights up a cigar. What is the probability that he is a liberal?
Five pennies are sitting on a table. One is a trick coin that has heads on both sides, but the other four are normal. You pick up a penny at random and flip it four times, getting heads each time. Given this, what is the probability that you picked up the two-headed penny?
One slot machine pays off 1/2 of the time, while another pays off 1/4 of the time. We pick one of the machines and play it 6 times, winning 3 times. What is the probability we are playing the machine that pays off only 1/4 of the time?
A student is taking a multiple-choice exam in which each question has four possible answers. She knows the answers to 60% of the questions and guesses at the others. What is the probability that she guessed given that she got question 12 right?
20% of people are “accident-prone” and have a probability 0.15 of having an accident in a 1-year period in contrast to a probability of 0.05 for the other 80% of people.(a) If we pick a person at random, what is the probability that they will have an accident this year? (b)What is the
One die has 4 red and 2 white sides; a second has 2 red and 4 white sides.(a) If we pick a die at random and roll it, what is the probability that the result is a red side?(b) If the first result is a red side and we roll the same die again, what is the probability of a second red side? (a) Let A
A particular football team is known to run 40% of its plays to the left and 60% to the right. When the play goes to the right, the right tackle shifts his stance 80% of the time, but does so only 10% of the time when the play goes to the left. As the team sets up for the play the right tackle
A company gives a test to 100 salesmen, 80 with good sales records and 20 with poor records. 60% of the good salesmen pass the test, but only 30% of the poor salesmen do. A new applicant takes the test and passes. What is the probability that he is a good salesman?
You are a serious student who studies on Friday nights but your roommate goes out and has a good time. 40% of the time he goes out with his girlfriend; 60% of the time he goes to a bar. 30% of the times when he goes out with his girlfriend he spends the night at her apartment. 40% of the times when
Two masked robbers try to rob a crowded bank during the lunch hour but the teller presses a button that sets off an alarm and locks the front door. The robbers, realizing they are trapped, throw away their masks and disappear into the chaotic crowd. Confronted with 40 people claiming they are
Three bags lie on the table. One has two gold coins, one has two silver coins, and one has one silver and one gold. You pick a bag at random, and pick out one coin. If this coin is gold, what is the probability that you picked from the bag with two gold coins?
In a certain city, 30% of the people are Conservatives, 50% are Liberals, and 20% are Independents. In a given election, 2/3 of the Conservatives voted, 80% of the Liberals voted, and 50% of the Independents voted. If we pick a voter at random what is the probability that she is Liberal?
An undergraduate student has asked a professor for a letter of recommendation. He estimates that the probability he will get the job is 0.8 with a strong letter, 0.4 with a medium letter, and 0.1 with a weak letter. He also believes that the probabilities that the letter will be strong, medium, or
A group of 20 people go out to dinner. 10 go to an Italian restaurant, 6 to a Japanese restaurant, and 4 to a French restaurant. The fractions of people satisfied with their meals were 4/5, 2/3, and 1/2 respectively. The next day the person you are talking to was satisfied with what they ate. What
1 out of 1,000 births results in fraternal twins; 1 out of 1,500 births results in identical twins. Identical twins must be the same sex but the sexes of fraternal twins are independent. If two girls are twins, what is the probability that they are fraternal twins?
Consider the following data on traffic accidents:Calculate.(a) The probability that a randomly chosen driver will have an accident this year,(b) The probability that a driver is between 46 and 65 given that they had an accident. Age group 16-25 26-45 46-65 >65 % of drivers % of drivers 15 35 35
Suppose we draw two tickets from a hat that contains tickets numbered 1, 2, 3, 4. Let X be the first number drawn and Y be the second. Find the joint distribution of X and Y.
Suppose we roll one die repeatedly and let Ni be the number of the roll on which i first appears. Find the joint distribution of N1 and N6.
Compute(a) P (X = 1|Y = 1)(b) P (X = 2|Y = 2)for the following joint distribution: Y 1 2 3 X = 1 0.1 0.15 0.05 2 0.2 0.15 0 3 0.3 0 0.05
Compute(a) P (X = 2|Y = 3)(b) P (Y = 3|X = 3)for the following joint distribution: Y 1 2 3 X = 1 0.2 0.10 0.05 2 0.15 0 0.15 3 0.05 0.10 0.20
Using the clues given below, fill in the rest of the joint distribution. There is only one answer. Y 3 1 ? ? 2 0.1 0.05 ? (a) P(Y=2|X = 0) = 1/4, (b) X and Y are independent. X = 0 6
Using the clues given below, fill in the rest of the joint distribution. There is only one answer: Y 1 X = 1 ? 2 3 ? 0 ? ? 0 2 3 0 For k= 1, 2, 3, (a) P(Y = 1|X = k) = 2/3, (b) P(X = k|Y = 1) = k/6.
Fill in the rest of the joint distribution so that X and Y are independent. There are two possible answers: Y X = 0 1 0 ? 2/9 2/9 ? 1 2.
Find the mean and variance of the number of games in the World Series.Recall that it is won by the first team to win four games and assume that the outcomes are determined by flipping a coin.
In a group of five items, two are defective. Find the distribution of N, the number of draws we need to find the first defective item. Find the mean and variance of N.
A die is rolled 8 times. What is the probability that we will get exactly two 3’s?
A random variable has P (X = x) = x/15 for x = 1, 2, 3, 4, 5, and 0 otherwise.Find the mean and variance of X.
Suppose X has density function f (x) = c(3 − |x|) when −3 < x < 3.What value of c makes this a density function?
Consider f (x) = c (1 − x2) for −1 < x < 1 and 0 otherwise. What value of c should we take to make f a density function?
Suppose X has density function 6x(1 − x) for 0 < x < 1 and 0 otherwise.Find(a) E X,(b) E (X2),(c) var (X).
Suppose X has density function x2/9 for 0 < x < 3 and 0 otherwise. Find(a) E X,(b) E (X2), (c) Var (X).
Suppose X has density function x−2/3/21 for 1 < x < 8 and 0 otherwise. Find(a) E X,(b) E (X2),(c) Var (X).
F (x) = 3x2 − 2x3 for 0 < x < 1 (with F (x) = 0 if x ≤ 0 and F (x) = 1 if x ≥ 1) defines a distribution function. Find the corresponding density function.
Let F (x) = e−1/x for x > 0 and F (x) = 0 for x ≤ 0. Is F a distribution function? If so, find its density function
Let F (x) = 3x − 2x2 for 0 ≤ x ≤ 1, F (x) = 0 for x ≤ 0, and F (x) = 1 for x ≥ 1. Is F a distribution function? If so, find its density function.
Suppose X has density function f (x) = x/2 for 0 < x < 2 and 0 otherwise. Find(a) The distribution function,(b) P (X < 1),(c) P (X > 3/2),(d) The median.
Suppose X has density function f (x) = 4x3 for 0 < x < 1 and 0 otherwise. Find(a) The distribution function,(b) P (X < 1/2),(c) P (1/3 < X < 2/3),(d) The median.
Suppose X has density function x−1/2/2 for 0 < x < 1 and0otherwise. Find(a) The distribution function,(b) P (X > 3/4),(c) P (1/9 < X < 1/4),(d) The median.
Suppose P (X = x) = x/21 for x = 1, 2, 3, 4, 5, 6. Find all the medians of this distribution.
Suppose X has a Poisson distribution with λ = ln 2. Find all the medians of X.

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