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elementary probability for applications
Elementary Probability For Applications 1st Edition Rick Durrett - Solutions
A basketball team has 5 players more than 6 feet tall and 6 who are less than 6 feet. How many ways can they have their picture taken if the 5 taller players stand in a row behind the 6 shorter players who are sitting on a row of chairs?
The Duke basketball team has 10 women who can play guard and 12 tall women who can play the other three positions. At the start of the game, the coach gives the referee a starting lineup that lists who will play left guard, right guard, left forward, center, and right forward. In how many ways can
Six students, three boys and three girls, lineup in a random order for a photograph. What is the probability that the boys and girls alternate?
Seven people sit at a round table. How many ways can this be done if Mr. Jones and Miss Smith(a) Must sit next to each other(b) Must not sit next to each other? (Two seating patterns that differ only by a rotation of the table are considered the same.)
How many ways can four rooks be put on a chessboard so that no rook can capture any other rook? Or, what is the same: How many ways can 8 markers be placed on an 8 × 8 grid of squares so that there is at most one in each row or column?
A BINGO card is a 5 × 5 grid. The center square is a free space and has no number. The first column is filled with five distinct numbers from 1 to 15, the second with five numbers from 16 to 30, the middle column with four numbers from 31 to 45, the fourth with five numbers from 46 to 60, and the
Continuing with the setup from the previous problem, in BINGO numbers are drawn from 1 to 75 without replacement. When a number is called you put a marker on that square. If you have five in a row horizontally, vertically, or diagonally, you have a BINGO. What is the probability you will have a
How many different ways can the letters in the following words be arranged:(a) Money.(b) Banana.(c) Statistics.(d) Mississippi?
12 different toys are to be divided among 3 children so that each one gets 4 toys. How many ways can this be done?
A club with 50 members is going to form two committees, one with 8 members and the other with 7. How many ways can this be done(a) If the committees must be disjoint?(b) If they can overlap?
If seven dice are rolled, what is the probability that each of the six numbers will appear at least once?
A die is rolled 8 times. What is the probability we will get exactly two 3’s?
Mary knows the answers to 20 of the 25 multiple choice questions on the Psychology 101 exam, but she has skipped several of the lectures, she must take random guesses for the other five. Assuming each question has four answers, what is the probability she will get exactly 3 of the last 5 questions
In 1997, 10.8% of female smokers smoked cigars. In a sample of size 10 female smokers What is the probability that(a) Exactly 2 of the women smoke cigars?(b) At most 1 smokes cigars.
A 1994 report revealed that 32.6% of U.S. births were to unmarried women. A parenting magazine selected 30 women who gave birth in 1994 at random.(a) What is the probability that exactly 10 of the women were unmarried?(b) Using your calculator determine the probability that in the sample at most
20% of all students are left-handed. A class of size 20 meets in a room with 5 left-handed and 18 right-handed chairs. Use your calculator to find the probability that each student will have a chair to match their needs.
David claims to be able to distinguish brand B beer from brand H but Alice claims that he just guesses. They set up a taste test with 10 small glasses of beer. David wins if he gets 8 or more right. What is the probability he will win(a) If he is just guessing?(b) If he gets the right answer with
The following situation comes up the game of Yahtzee. We have three rolls of five dice and want to get three sixes or more. On each turn we reroll any dice that are not 6’s. What is the probability we succeed?
A baseball pitcher throws a strike with probability 0.5 and a ball with probability 0.5. He is facing a batter who never swings at a pitch. What is the probability that he strikes out, i.e., gets three strikes before four balls?
A baseball player is said to “hit for the cycle” if he has a single, a double, a triple, and a home run all in one game. Suppose these four types of hits have probabilities 1/6, 1/20, 1/120, and 1/24. What is the probability of hitting for the cycle if he gets to bat (a) four times, (b) five
Compare the Poisson approximation with the exact binomial probabilities when(a) N = 10, p = 0.1,(b) N = 20, p = 0.05,(c) N = 40, p = 0.025.
Use the Poisson approximation to compute the probability that you will roll at least one double 6 in 24 trials. How does this compare with the exact answer?
The probability of a three of a kind in poker is approximately 1/50. Use the Poisson approximation to compute the probability you will get at least one three of a kind if you play 20 hands of poker.
Calls to a toll-free hotline service are made randomly at rate two per minute. The service has five operators, none of whom is currently busy. Use the Poisson distribution to estimate the probability that in the next minute there are < 5 calls.
In one of the New York state lottery games, a number is chosen at random between 0 and 999. Suppose you play this game 250 times. Use the Poisson approximation to estimate the probability you will never win and compare this with the exact answer.
If you bet $1 on number 13 at roulette (or on any other number) then you win $35 if that number comes up, an event of probability 1/38, and you lose your dollar otherwise. Suppose you play 70 times. Use the Poisson approximation estimate the probability that(a) You have won 0 times and lost
In a particular Powerball drawing 210,850,582 tickets were sold. The chance of winning the lottery is 1 in 80,000,000. Use the Poisson approximation to estimate the probability that there is exactly one winner.
Suppose that the probability of a defect in a foot of magnetic tape is 0.002. Use the Poisson approximation to compute the probability that a 1500 foot roll will have no defects.
Suppose 1% of a certain brand of Christmas lights is defective. Use the Poisson approximation to compute the probability that in a box of 25 there will be at most one defective bulb.
In February 2000, 2.8% of Colorado’s labor force was unemployed. Calculate the probability that in a group of 50 workers exactly one is unemployed.
An insurance company insures 3000 people, each of whom has a 1/1000 chance of an accident in one year. Use the Poisson approximation to compute the probability there will be at most 2 accidents.
Suppose that 1% of people in the population are over 6 feet 3 inches tall. What is the chance that in a group of 200 people picked at random from the population at least four people will be over 6 feet 3 inches tall.
An airline company sells 160 tickets for a plane with 150 seats, knowing that the probability that a passenger will not show up for the flight is 0.1. Use the Poisson approximation to compute the probability that they will have enough seats for all the passengers who show up.
An airline company sells 160 tickets for a plane with 150 seats, knowing that the probability a passenger will not show up for the flight is 0.1. Use the Poisson approximation to compute the probability they will have enough seats for all the passengers who show up.
Books from a certain publisher contain an average of 1 misprint per page. What is the probability that on at least one page in a 300 page book there are five misprints?
Two red cards and two black cards are lying face down on the table. You pick two cards and turn them over. What is the probability that the two cards are different colors?
Four people are chosen at random from 5 couples. What is the probability two men and two women are selected?
You pick 5 cards out of a deck of 52. What is the probability you get exactly 2 spades?
Seven students are chosen at random from a class with 17 boys and 13 girls. What is the probability that 4 boys and 3 girls are selected?
In a carton of 12 eggs, 2 are rotten. If we pick 4 eggs to make an omelet, what is the probability we do not get a rotten egg?
An electronics store receives a shipment of 30 calculators of which 4 are defective. Six of these calculators are selected to be sent to a local high school. What is the probability that exactly one is defective?
A scrabble set contains 54 consonants, 44 vowels, and 2 blank tiles. Find the probability that your initial draw contains 5 consonants and 2 vowels.
(a) How many ways can can we pick 4 students from a group of 30 to be on the math team?(b) Suppose there are 18 boys and 12 girls. What is the probability the team will have 2 boys and 2 girls.
The following probability problem arose in a court case concerning possible discrimination against black nurses. 26 white nurses and 9 black nurses took an exam. All the white nurses passed but only 4 of the black nurses did. What is the probability we would get this outcome if the five nurses who
A closet contains 8 pairs of shoes. You pick out 5. What is the probability of(a) No pair,(b) Exactly one pair,(c) Two pairs?
A drawer contains 10 black, 8 brown, and 6 blue socks. If we pick two socks at random, what is the probability they match?
A dance class consists of 12 men and 10 women. Five men and five women are chosen and paired up to dance. In how many ways can this be done?
Suppose we pick 5 cards out of a deck of 52. What is the probability we get at least one card of each suit?
A bridge hand in which there is no card higher than a 9 is called a Yarborough after the Earl who liked to bet at 1000 to 1 that your bridge hand would have a card that was 10 or higher. What is the probability of a Yarborough when you draw 13 cards out of a deck of 52.
Two cards are a blackjack if one is an A and the other is a K, Q, J, or 10.(a) If you pick two cards out of a deck, what is the probability you will get a blackjack?(b) Suppose you are playing blackjack against the dealer with a freshly shuffled deck. What is the probability that you or the
A student studies 12 problems from which the professor will randomly choose 6 for a test. If the student can solve 9 of the problems, what is the probability she can solve at least 5 of the problems on the test?
A football team has 16 seniors, 12 juniors, 8 sophomores, and 4 freshmen.(a) If we pick 5 players at random, what is the probability we will get 2 seniors and 1 from each of the other 3 classes?(b) What is the most likely outcome?
In a kindergarten class of 20 students, one child is picked each day to help serve the morning snack. What is the probability that in one week five different children are chosen?
An investor picks 3 stocks out of 10 recommended by his broker. Of these, six will show a profit in the next year. What is the probability the investor will pick(a) 3(b) 2(c) 1(d) 0 profitable stocks?
Four red cards (i.e., hearts and diamonds) and four black cards are face down on the table. A psychic who claims to be able to locate the four black cards turns over 4 cards and gets 3 black cards and 1 red card. What is the probability he would get 3 or 4 black cards if he were guessing?
A town council considers the question of closing down an “adult” theatre.The five men on the council all vote against this and the three women vote in favor. What is the probability we would get this result(a) If the council members determined their votes by flipping a coin?(b) If we assigned
An urn contains white balls numbered 1 to 15 and black balls also numbered 1 to 15. Suppose you draw 4 balls. What is the probability that(a) No two have the same number?(b) You get exactly one pair with the same number?(c) You get two pair with the same numbers?
A town has four TV repairmen. In the first week of September four TV sets break and their owners call repairmen chosen at random. Find the probability that the number of repairmen who have jobs is 1, 2, 3, 4.
Compute the probabilities of the following poker hands when we roll five six sided dice. (a) five of a kind (b) four of a kind, (c) a full house, (d) three of a kind (e) two pair (f) one pair (g) no pair 0.000771 0.019290 0.038580 0.154320 0.231481 0.462962 0.092592
In seven-card stud you receive seven cards and use them to make the best poker hand you can. Ignoring the possibility of a straight or a flush the probability that the best hand you can make with your cards isVerify the probabilities for seven card stud. Hint: For full house you need to consider
Six high school teams play each other in the Southern Tier division. Each team plays all of the other teams once. What is the probability some team has a perfect 5 − 0 season?
Suppose you draw seven cards out of a deck of 52. What is the probability you will have(a) Exactly five cards of one suit?(b) At least five cards of one suit?
In a certain city 60% of the people subscribe to newspaper A, 50% to B, 40% to C, 30% to A and B, 20% to B and C, and 10% to A and C, but no one subscribes to all three. What percentage subscribe to(a) At least one newspaper?(b) Exactly one newspaper?(c) Only to A?(d) Only to B?
Santa Claus has 45 drums, 50 cars, and 55 baseball bats in his sled. 15 boys will get a drum and a car, 20 a drum and a bat, 25 a bat and a car, and 5 will get three presents.(a) How many boys will receive presents?(b) How many boys will get just a drum?
Use the inclusion-exclusion formula to compute the probability that a randomly chosen number between 0000 and 9999 contains at least one 1. Check this by computing the probability there is no 1.
Ten people call an electrician and ask him to come to their houses on randomly chosen days of the work week (Monday through Friday). What is the probability of A = “he has at least one day with no jobs”?
We pick a number between 0 and 999, then a computer picks one at random from that range. Use (2.15) to compute the probability at least two of our digits will match the computer’s number. (Note: We include any leading zeros, so 017 and 057 have two matching digits.)
You pick 13 cards out of a deck of 52. What is the probability that you will not get a card from every suit?
You pick 13 cards out of a deck of 52. Let A = “you have exactly six cards in at least one suit” and B = “you have exactly six spades.” The first Bonferroni inequality says that P (A) ≤ 4P(B). Compute P (A) and P(A) /P(B).
Use the first two Bonferroni inequalities to compute an upper and a lower bound on the probability that in a group of 60 people, at least 3 were born on the same day.
Suppose we roll two dice 6 times. Use the first three Bonferroni inequalities to compute bounds on the probability of A = “we get at least one double 6.” Compare the bounds with the exact answer 1 − (35/36)6.
Suppose we try 20 times for an event with probability 0.01. Use the first three Bonferroni inequalities to compute bounds on the probability of one success.Suppose we try 20 times for an event with probability 0.01. Use the first three Bonferroni inequalities to compute bounds on the probability of
A friend flips two coins and tells you that at least one is head. Given this information, what is the probability that the first coin is head?
A friend rolls two dice and tells you that there is at least one 6. What is the probability the sum is at least 9?
Suppose we roll two dice. What is the probability that the sum is 7 given that neither die showed a 6?
Suppose you draw 5 cards out of a deck of 52 and get 2 spades and 3 hearts.What is the probability that the first card drawn was a spade?
Two people, whom we call South and North, draw 13 cards out of a deck of 52. South has two aces. What is the probability that North has(a) None?(b) One?(c) The other two?
An urn contains 8 red, 7 blue, and 5 green balls. You draw out two balls and they are different colors. Given this, what is the probability that the two balls were red and blue?
Suppose 60% of the people subscribe to newspaper A, 40% to newspaper B, and 30% to both. If we pick a person at random who subscribes to at least one newspaper, what is the probability that she subscribes to newspaper A?
In a town 40% of families have a dog and 30% have a cat. 25% of families with a dog also have a cat.(a) What fraction of people have a dog or cat?(b) What is the probability that a family with a cat has a dog?
Plumber Bob does 40% of the plumbing jobs in a small town. 30% of the people in town are unhappy with their plumbers, but 50% of Bob’s customers are unhappy with his work. If your neighbor is not happy with his plumber, what is the probability that it was Bob?
An ectopic pregnancy is twice as likely if a woman smokes cigarettes. If 25% of women of childbearing age are smokers, what fraction of ectopic pregnancies occur to smokers?
Brown eyes are dominant over blue. That is, there are two alleles B and b. bb individuals have blue eyes but other combinations have brown eyes. Your parents and you have brown eyes but your brother has blue. So you can infer that both of your parents are heterozygotes, that is, have genetic type
Suppose that the probability that a married man votes is 0.45, the probability a married woman votes is 0.4, and the probability a woman votes given that her husband does is 0.6.What is the probability that(a) Both vote?(b) A man votes given that his wife does?
Two events have P (A) = 1/4, P (B|A) = 1/2, and P (A|B) = 1/3.Compute P (A ∩ B), P (B), and P (A ∪ B).
From a signpost that says MIAMI two letters fall off. A friendly drunk puts the two letters back into the two empty slots at random. What is the probability that the sign still says MIAMI?
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 she will correctly answer a
A man receives presents from his three children, Allison, Betty, and Chelsea.To avoid disputes he opens the presents in a random order. What are the possible outcomes?
Suppose we pick a number at random from the phone book and look at the last digit.(a) What is the set of outcomes and what probability should be assigned to each outcome? (b) Would this model be appropriate if we were looking at the first digit?
Two students arrive late for a math final exam with the excuse that their car had a flat tire. Suspicious, the professor says “each one of you write down on a piece of paper which tire was flat.” What is the probability that both students pick the same tire?
Suppose we roll a red die and a green die. What is the probability that the number on the red die is larger than the number on the green die?
Two dice are rolled. What is the probability that (a) the two numbers will differ by 1 or less and (b) the maximum of the two numbers will be 5 or larger?
If we flip a coin 5 times, what is the probability that the number of heads is an even number (that is, divisible by 2)?
The 1987 World Series was tied at two games a piece before the St. Louis Cardinals won the fifth game. According to the Associated Press, “The numbers of history support the Cardinals and the momentum they carry. Whenever the series has been tied 2–2 the team that won the fifth game won the
Two boys are repeatedly playing a game that they each have probability 1/2 of winning. The first person to win 5 games wins the match. What is the probability that Al will win if(a) He has won 4 games and Bobby has won 3 and(b) He leads by a score of 3 games to 2?
20 families live in a neighborhood: 4 have 1 child, 8 have 2 children, 5 have 3 children, and 3 have 4 children. If we pick a child at random, what is the probability that they come from a family with 1, 2, 3, 4 children?
In Galileo’s time people thought that when three dice were rolled, a sum of 9 and a sum of 10 had the same probability since each could be obtained in 6 ways:Compute the probabilities of these sums and show that 10 is a more likely total than 9. 9: 1+2 +6, 1+3+5, 1+4+4, 2+2+5, 2+3+4, 3+3+3 10:
Suppose we roll three dice. Compute the probability that the sum is (a) 3,(b) 4, (c) 5, (d) 6, (e) 7, (f) 8, (g) 9, (h) 10.
In a group of students, 25% smoke cigarettes, 60% drink alcohol, and 15% do both. What fraction of students have at least one of these bad habits?
In a group of 320 high school graduates, only 160 went to college but 100 of the 170 men did. How many women did not go to college?
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