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A First Course In Probability 9th Edition Sheldon Ross - Solutions

Suppose we have 10 coins such that if the ith coin is flipped, heads will appear with probability i/10, i = 1, 2, . . . , 10. When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin?
In any given year, a male automobile policyholder will make a claim with probability pm and a female policyholder will make a claim with probability pf, where pf ≠ pm. The fraction of the policyholders that are male is α, 0 < α < 1. A policyholder is randomly chosen. If Ai denotes the
An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is randomly chosen from the urn. What is the probability that all of the balls selected are white? What is the conditional probability that the die landed on 3 if all the balls selected are white?
Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the
Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the prostate-specific antigen (PSA) that is produced only by the prostate gland. Although PSA levels are indicative of
Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. The company’s records indicate that the probabilities that good-, average-, and bad-risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15,
A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she
A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities of receiving notification on each day of next week, given that she is accepted and that she is rejected, are as follows:She
A parallel system functions whenever at least one of its components works. Consider a parallel system of n components, and suppose that each component works independently with probability 1/2. Find the conditional probability that component 1 works given that the system is functioning.
In a class, there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?
A simplified model for the movement of the price of a stock supposes that on each day the stock’s price either moves up 1 unit with probability p or moves down 1 unit with probability 1 − p. The changes on different days are assumed to be independent.(a) What is the probability that after 2
Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin which lands on heads with some unknown probability p that need not be equal to 1/2. Consider the following procedure for accomplishing our task:1. Flip the coin.2. Flip the
Independent flips of a coin that lands on heads with probability p are made. What is the probability that the first four outcomes are(a) H, H, H, H?(b) T, H, H, H?(c) What is the probability that the pattern T, H, H, H occurs before the pattern H, H, H, H? How can the pattern H, H, H, H occur first?
The color of a person’s eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person will have brown eyes; and if one of them is a blue-eyed gene and the other a brown-eyed gene, then the
Genes relating to albinism are denoted by A and a. Only those people who receive the a gene from both parents will be albino. Persons having the gene pair A, a are normal in appearance and, because they can pass on the trait to their offspring, are called carriers. Suppose that a normal couple has
Barbara and Dianne go target shooting. Suppose that each of Barbara’s shots hits a wooden duck target with probability p1, while each shot of Dianne’s hits it with probability p2. Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it
A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each
A true–false question is to be posed to a husband-and-wife team on a quiz show. Both the husband and the wife will independently give the correct answer with probability p. Which of the following is a better strategy for the couple? (a) Choose one of them and let that person answer the
The probability of the closing of the ith relay in the circuits shown in Figure 3.4 is given by pi, i = 1, 2, 3, 4, 5. If all relays function independently, what is the probability that a current flows between A and B for the respective circuits?Condition on whether relay 3 closes.(a)(b)FIGURE 3.4:
In Problem 3.66a, find the conditional probability that relays 1 and 2 are both closed given that a current flows from A to B.Problem 3.66The probability of the closing of the ith relay in the circuits shown in Figure 3.4 is given by pi, i = 1, 2, 3, 4, 5. If all relays function independently, what
A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense
There is a 50–50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50–50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is a fourth prince, what
On the morning of September 30, 1982, the won€“lost records of the three leading baseball teams in the Western Division of the National League were as follows:Each team had 3 games remaining. All 3 of the Giants€™ games were with the Dodgers, and the 3 remaining games of the
A town council of 7 members contains a steering committee of size 3. New ideas for legislation go first to the steering committee and then on to the council as a whole if at least 2 of the 3 committee members approve the legislation. Once at the full council, the legislation requires a majority
Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having 5 children, compute the probabilities of the following events: (a) All children are of the same sex. (b) The 3 eldest are
A and B alternate rolling a pair of dice, stopping either when A rolls the sum 9 or when B rolls the sum 6. Assuming that A rolls first, find the probability that the final roll is made by A.
In a certain village, it is traditional for the eldest son (or the older son in a two-son family) and his wife to be responsible for taking care of his parents as they age. In recent years, however, the women of this village, not wanting that responsibility, have not looked favorably upon marrying
Suppose that E and F are mutually exclusive events of an experiment. Show that if independent trials of this experiment are performed, then E will occur before F with probability P(E)/[P(E) + P(F)].
Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes 1, 2, or 3. Given that outcome 3 is the last of the three outcomes to occur, find the conditional probability that(a) The first trial results in outcome 1;(b) The first two
A and B play a series of games. Each game is independently won by A with probability p and by B with probability 1 − p. They stop when the total number of wins of one of the players is two greater than that of the other player. The player with the greater number of total wins is declared the
In successive rolls of a pair of fair dice, what is the probability of getting 2 sevens before 6 even numbers?
In a certain contest, the players are of equal skill and the probability is 1/2 that a specified one of the two contestants will be the victor. In a group of 2n players, the players are paired off against each other at random. The 2n−1 winners are again paired off randomly, and so on, until a
A and B flip coins. A starts and continues flipping until a tail occurs, at which point B starts flipping and continues until there is a tail. Then A takes over, and so on. Let P1 be the probability of the coin’s landing on heads when A flips and P2 when B flips. The winner of the game is the
Die A has 4 red and 2 white faces, whereas die B has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game continues with die A; if it lands on tails, then die B is to be used.(a) Show that the probability of red at any throw is 1/2.(b) If the first two throws result
Let S = {1, 2, . . . , n} and suppose that A and B are, independently, equally likely to be any of the 2n subsets (including the null set and S itself) of S.(a) Show thatLet N(B) denote the number of elements in B. Use(b) Show that
Consider 3 urns. Urn A contains 2 white and 4 red balls, urn B contains 8 white and 4 red balls, and urn C contains 1 white and 3 red balls. If 1 ball is selected from each urn, what is the probability that the ball chosen from urn A was white given that exactly 2 white balls were selected?
A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability .7, whereas when the defendant is in fact innocent, this probability drops to .2. If 70
Suppose that n independent trials, each of which results in any of the outcomes 0, 1, or 2, with respective probabilities p0, p1, and p2,are performed. Find the probability that outcomes 1 and 2 both occur at least once.
Consider a collection of n individuals. Assume that each person’s birthday is equally likely to be any of the 365 days of the year and also that the birthdays are independent. Let Ai,j, i ≠ j, denote the event that persons i and j have the same birthday. Show that these events are pairwise
Show that 0 ‰¤ ai ‰¤ 1, i = 1, 2, . . ., thenSuppose that an infinite number of coins are to be flipped. Let ai be the probability that the ith coin lands on heads, and consider when the first head occurs.
The probability of getting a head on a single toss of a coin is p. Suppose that A starts and continues to flip the coin until a tail shows up, at which point B starts flipping. Then B continues to flip until a tail comes up, at which point A takes over, and so on. Let Pn,m denote the probability
Independent trials that result in a success with probability p are successively performed until a total of r successes is obtained. Show that the probability that exactly n trials are required isUse this result to solve the problem of the points (Example 4j).In order for it to take n trials to
Independent trials that result in a success with probability p and a failure with probability 1 − p are called Bernoulli trials. Let Pn denote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show thatPn = p(1 − Pn−1) + (1 −
Let A ⊂ B. Express the following probabilities as simply as possible: P(A|B), P(A|Bc), P(B|A), P(B|Ac)
As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. Show that the weather is dry on January 1, then Pn, the probability that it will be dry n days later, satisfies Pn = (2p − 1)Pn−1 + (1
A bag contains a white and b black balls. Balls are chosen from the bag according to the following method:1. A ball is chosen at random and is discarded.2. A second ball is then chosen. If its color is different from that of the preceding ball, it is replaced in the bag and the process is repeated
A round-robin tournament of n contestants is a tournament in which each of thepairs of contestants play each other exactly once, with the outcome of any play being that one of the contestants wins and the other loses. For a fixed integer k, k then such an outcome is possible.Suppose that the
Prove directly that P(E|F) = P(E|FG)P(G|F) + P(E|FGc)P(Gc|F)
Extend the definition of conditional independence to more than 2 events.
In Laplace’s rule of succession (Example 5e), show that if the first n flips all result in heads, then the conditional probability that the next m flips also result in all heads is (n + 1)/(n + m + 1).
Consider a school community of m families, with ni of them having i children,Consider the following two methods for choosing a child:1. Choose one of the m families at random and then randomly choose a child from that family.2. Choose one of theini children at random.Show that method 1 is more
A ball is in any one of n boxes and is in the ith box with probability Pi. If the ball is in box i, a search of that box will uncover it with probability Î±i. Show that the conditional probability that the ball is in box j, given that a search of box i did not uncover it, is
Prove that if E1, E2, . . . ,En are independent events, then
(a) An urn contains n white and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that, with probability n/(n + m), they are all white.Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn.(b)
Let A,B, and C be events relating to the experiment of rolling a pair of dice.(a) If P(A|C) > P(B|C) and P(A|Cc) > P(B|Cc) either prove that P(A) > P(B) or give a counterexample by defining events A,B, and C for which that relationship is not true.(b) If P(A|C) > P(A|Cc) and P(B|C) >
Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with
(a) An integer N is to be selected at random from {1, 2, . . . , (10)3} in the sense that each integer has the same probability of being selected. What is the probability that N will be divisible by 3? by 5? by 7? by 15? by 105? How would your answer change if (10)3 is replaced by (10)k as k became
A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .3, and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard
Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares her number with that of player 3, and so on. Let
The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won–lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have
In Problem 15, let team number 1 be the team with the worst record, let team number 2 be the team with the second-worst record, and so on. Let Yi denote the team that gets draft pick number i. (Thus, Y1 = 3 if the first ball chosen belongs to team number 3.) Find the probability mass function
Two fair dice are rolled. Let X equal the product of the 2 dice. Compute P{X = i} for i = 1, . . . , 36.
A gambling book recommends the following “winning strategy” for the game of roulette: Bet $1 on red. If red appears (which has probability 18/38), then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability 20/38 of occurring), make additional $1 bets
Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus
Suppose that two teams play a series of games that ends when one of them has won i games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when (a) i = 2 and (b) i = 3. Also, show in both cases that this number is
You have $1000, and a certain commodity presently sells for $2 per ounce. Suppose that after one week the commodity will sell for either $1 or $4 an ounce, with these two possibilities being equally likely.(a) If your objective is to maximize the expected amount of money that you possess at the end
A and B play the following game: A writes down either number 1 or number 2, and B must guess which one. If the number that A has written down is i and B has guessed correctly, B receives i units from A. If B makes a wrong guess, B pays 3/4 unit to A. If B randomizes his decision by guessing 1 with
Two coins are to be flipped. The first coin will land on heads with probability .6, the second with probability .7. Assume that the results of the flips are independent, and let X equal the total number of heads that result. (a) Find P{X = 1}. (b) Determine E[X].
There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p
A person tosses a fair coin until a tail appears for the first time. If the tail appears on the nth flip, the person wins 2n dollars. Let X denote the player’s winnings. Show that E[X] = +∞. This problem is known as the St. Petersburg paradox.(a) Would you be willing to pay $1 million to play
Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of 1 − (1 − p)2 if it
To determine whether they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of 10. The blood samples of the 10 people in each group will be pooled and analyzed
A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $1.10; if they are different colors, then you win −$1.00. (That is, you lose $1.00.) Calculate (a) The expected value of the amount you win; (b) The variance of the amount you win.
Consider Problem 22 with i = 2. Find the variance of the number of games played, and show that this number is maximized when p = 1/2. Problem 22 Suppose that two teams play a series of games that ends when one of them has won i games. Suppose that each game played is, independently, won by team A
Find Var(X) and Var(Y) for X and Y as given in Problem 21. Problem 21 Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students that
If E[X] = 1 and Var(X) = 5, find (a) E[(2 + X)2]; (b) Var(4 + 3X).
Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman. (For instance, X = 1 if the top-ranked person is female.) Find P{X = i}, i = 1, 2,
A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability p1, whereas on a dry day they each independently function with probability p2. If the
A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on day, then each of his examiners will pass him, independently of each other, with probability .8, whereas if he has
Suppose that it takes at least 9 votes from a 12-member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is .2, whereas the probability that the juror votes an innocent person guilty is .1. If each juror acts independently and if 65 percent
In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of
It is known that diskettes produced by a certain company will be defective with probability .01, independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective. The guarantee
When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times. (a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips? (b) Given that the first
Suppose that a biased coin that lands on heads with probability p is flipped 10 times. Given that a total of 6 heads results, find the conditional probability that the first 3 outcomes are(a) h, t, t (meaning that the first flip results in heads, the second in tails, and the third in tails);(b) t,
The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that the next page you read contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning!
The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be(a) At least 2 such accidents in the next month;(b) At most 1 accident in the next month? Explain your reasoning!
Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that, for at least one of these couples,(a) Both partners were born on April 30;(b) Both partners celebrated their birthday on the same day of the year.State your assumptions.
How many people are needed so that the probability that at least one of them has the same birthday as you is greater than 1/2?
Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ = 3. (a) Find the probability that 3 or more accidents occur today. (b) Repeat part (a) under the assumption that at least 1 accident occurs today.
If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100, what is the (approximate) probability that you will win a prize (a) At least once? (b) Exactly once? (c) At least twice?
In Problem 5, for n = 3, if the coin is assumed fair, what are the probabilities associated with the values that X can take on? Problem 5 Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?
The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter λ = 5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ = 3 for 75 percent of the population. For
Consider n independent trials, each of which results in one of the outcomes 1, . . . , k with respective probabilitiesShow that if all the pi are small, then the probability that no trial outcome occurs more than once is approximately equal to
People enter a gambling casino at a rate of 1 every 2 minutes. (a) What is the probability that no one enters between 12:00 and 12:05? (b) What is the probability that at least 4 people enter the casino during that time?
The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. (a) Find the probability that, in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month. (b) What is the probability that there will be at least 2 months during the year
Each of 500 soldiers in an army company independently has a certain disease with probability 1/103. This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested. (a) What is the (approximate) probability that the blood test will be
A total of 2n people, consisting of n married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let Ci denote the event that the members of couple i are seated next to each other, i = 1, . . . , n. (a) Find P(Ci). (b) For j ≠ i, find P(Cj|Ci). (c)
Repeat the preceding problem when the seating is random but subject to the constraint that the men and women alternate. Preceding problem A total of 2n people, consisting of n married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let Ci denote the
Suppose in Problem 72 that the two teams are evenly matched and each has probability 12 of winning each game. Find the expected number of games played.Problem 72Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is
An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be interviewed with probability 2/3, what is the probability that her list of people will enable her to obtain her necessary number of interviews

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