Chapter 3

Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?

If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is \(i ?\) Compute for all values of \(i\) between 2 and 12 .

What is the probability that at least one of a pair of fair dice lands on \(6,\) given that the sum of the dice is \(i\) \(i=2,3, \ldots, 12 ?\)

An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the first 2 selected are white and the last 2 black?

A couple has 2 children. What is the probability that both are girls if the older of the two is a girl?

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?

Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades.

Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let \(B\) be the event that both cards are aces, let \(A_{s}\) be the event that the ace of spades is chosen, and let \(A\) be the event that at least one ace is chosen. Find (a) \(P\left(B | A_{s}\right)\) (b) \(P(B | A)\)

A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is \(.9 .\) If she passes the first exam, then the conditional probability that she passes the second one is \(.8,\) and if she passes both the first and the second exams, then the conditional probability that she passes the third exam is .7. (a) What is the probability that she passes all three exams? (b) Given that she did not pass all three exams, what is the conditional probability that she failed the second exam?

Suppose that an ordinary deck of 52 cards (which contains \(4 \text { aces })\) is randomly divided into 4 hands of 13 cards each. We are interested in determining \(p,\) the probability that each hand has an ace. Let \(E_{i}\) be the event that the \(i\) th hand has exactly one ace. Determine \(p=\) \(P\left(E_{1} E_{2} E_{3} E_{4}\right)\) by using the multiplication rule.

An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that (a) the first 2 balls selected are black and the next 2 are white; (b) of the first 4 balls selected, exactly 2 are black.

Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed, the baby survives 96 percent of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?

In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is (a) the probability that a randomly selected family owns both a dog and a cat? (b) the conditional probability that a randomly selected family owns a dog given that it owns a cat?

A total of 46 percent of the voters in a certain city classify themselves as Independents, whereas 30 percent classify themselves as Liberals and 24 percent say that they are Conservatives. In a recent local election, 35 percent of the Independents, 62 percent of the Liberals, and 58 percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is (a) an Independent? (b) a Liberal? (c) a Conservative? (d) What percent of voters participated in the local election?

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) the student is female given that the student is majoring in computer science; (b) this student is majoring in computer science given that the student is female.

A red die, a blue die, and a yellow die (all six sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. That is, with \(B, Y,\) and \(R\) denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in \(P(B

Urn I contains 2 white and 4 red balls, whereas urn II contains 1 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. What is (a) the probability that the ball selected from urn II is white? (b) the conditional probability that the transferred ball was white given that a white ball is selected from urn II?

3.24. Each of 2 balls is painted either black or gold and then placed in an urn. Suppose that each ball is colored black with probability \(\frac{1}{2}\) and that these events are independent. (a) Suppose that you obtain information that the gold paint has been used (and thus at least one of the balls is painted gold). Compute the conditional probability that both balls are painted gold. (b) Suppose now that the urn tips over and 1 ball falls out. It is painted gold. What is the probability that both balls are gold in this case? Explain.

3.25. The following method was proposed to estimate the number of people over the age of 50 who reside in a town of known population 100,000: "As you walk along the streets, keep a running count of the percentage of people you encounter who are over \(50 .\) Do this for a few days; then multiply the percentage you obtain by 100,000 to obtain the estimate." Comment on this method. Hint: Let \(p\) denote the proportion of people in the town who are over \(50 .\) Furthermore, let \(\alpha_{1}\) denote the proportion of time that a person under the age of 50 spends in the streets, and let \(\alpha_{2}\) be the corresponding value for those over \(50 .\) What quantity does the method suggested estimate? When is the estimate approximately equal to \(p ?\)

Suppose that 5 percent of men and 0.25 percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

All the workers at a certain company drive to work and park in the company's lot. The company is interested in estimating the average number of workers in a car. Which of the following methods will enable the company to estimate this quantity? Explain your answer. 1\. Randomly choose \(n\) workers, find out how many were in the cars in which they were driven, and take the average of the \(n\) values. 2\. Randomly choose \(n\) cars in the lot, find out how many were driven in those cars, and take the average of the \(n\) values.

Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional probability that the card following it is the (a) ace of spades? (b) two of clubs?

There are 15 tennis balls in a box, of which 9 have not previously been used. Three of the balls are randomly chosen, played with, and then returned to the box. Later, another 3 balls are randomly chosen from the box. Find the probability that none of these balls has ever been used.

Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. \(A\) box is selected at random, and a marble is drawn from it at random. What is the probability that the marble is black? What is the probability that the first box was the one selected given that the marble is white?

Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the news is bad. So, being a student of probability, Ms. Aquina instructs the doctor to flip a coin. If it comes up heads, the doctor is to call if the news is good and not call if the news is bad. If the coin comes up tails, the doctor is not to call. In this way, even if the doctor doesn't call, the news is not necessarily bad. Let \(\alpha\) be the probability that the tumor is cancerous; let \(\beta\) be the conditional probability that the tumor is cancerous given that the doctor does not call. (a) Which should be larger, \(\alpha\) or \(\beta ?\) (b) Find \(\beta\) in terms of \(\alpha,\) and prove your answer in part (a). 3.32. A family has \(j\) children with probability \(p_{j},\) where \(p_{1}=.1, p_{2}=.25, p_{3}=.35, p_{4}=.3 .\) A child from this fam- ily is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has (a) only 1 child; (b) 4 children. Redo (a) and (b) when the randomly selected child is the youngest child of the family.

On rainy days, Joe is late to work with probability \(.3 ;\) on nonrainy days, he is late with probability.1. With probability.7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?

Stores \(A, B,\) and \(C\) have \(50,75,\) and 100 employees, respectively, and \(50,60,\) and 70 percent of them respectively are women. Resignations are equally likely among all employees, regardless of sex. One woman employee resigns. What is the probability that she works in store \(C ?\)

Urn \(A\) has 5 white and 7 black balls. Urn \(B\) has 3 white and 12 black balls. We flip a fair coin. If the outcome is heads, then a ball from urn \(A\) is selected, whereas if the outcome is tails, then a ball from urn \(B\) is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and 7 red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color. Find the probability that the sample will contain exactly (a) 0 white balls; (b) 1 white ball; (c) 3 white balls; (d) 2 white balls.

A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled, and a card is drawn from it. Compute the probability that this drawn card is an ace. Hint: Condition on whether or not the interchanged card is selected.

Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \(\$ 250,000\) per year, while a total of 3.3 percent of all California households earn more than \(\$ 250,000\) per year. (a) What proportion of all non-California households earn more than \(\$ 250,000\) per year? (b) Given that a randomly chosen U.S. household earns more than \(\$ 250,000\) per year, what is the probability it is a California household?

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner \(A\) asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if \(A\) knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from \(\frac{1}{3}\) to \(\frac{1}{2}\) because he would then be one of two prisoners. What do you think of the jailer's reasoning?

Suppose we have 10 coins such that if the \(i\) th coin is flipped, heads will appear with probability \(i / 10, i=\) \(1,2, \ldots, 10 .\) 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 \(p_{m}\) and a female policyholder will make a claim with probability \(p_{f},\) where \(p_{f} \neq p_{m} .\) The fraction of the policyholders that are male is \(\alpha, 0<\alpha<1 .\) A policyholder is randomly chosen. If \(A_{i}\) denotes the event that this policyholder will make a claim in year \(i,\) show that $$ P\left(A_{2} | A_{1}\right)>P\left(A_{1}\right) $$ Give an intuitive explanation of why the preceding inequality is true.

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 probability that there is a silver coin in the other drawer?

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 cancer, the test is notoriously unreliable. Indeed, the probability that a noncancerous man will have an elevated PSA level is approximately. \(135,\) increasing to approximately.268 if the man does have cancer. If, on the basis of other factors, a physician is 70 percent certain that a male has prostate cancer, what is the conditional probability that he has the cancer given that (a) the test indicated an elevated PSA level? (b) the test did not indicate an elevated PSA level? Repeat the preceding calculation, this time assuming that the physician initially believes that there is a 30 percent chance that the man has prostate cancer.

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 and \(30 .\) If 20 percent of the population is a good risk, 50 percent an average risk, and 30 percent a bad risk, what proportion of people have accidents in a fixed year? If policyholder \(A\) had no accidents in \(2012,\) what is the probability that he or she is a good risk? is an average risk?

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 receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, and weak are .7 \(.2,\) and \(.1,\) respectively. (a) How certain is she that she will receive the new job offer? (b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation? (c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?

If you had to construct a mathematical model for events \(E\) and \(F,\) as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning. (a) \(E\) is the event that a businesswoman has blue eyes, and \(F\) is the event that her secretary has blue eyes. (b) \(E\) is the event that a professor owns a car, and \(F\) is the event that he is listed in the telephone book. (c) \(E\) is the event that a man is under 6 feet tall, and \(F\) is the event that he weighs more than 200 pounds. (d) \(E\) is the event that a woman lives in the United States, and \(F\) is the event that she lives in the Western Hemisphere. (e) \(E\) is the event that it will rain tomorrow, and \(F\) is the event that it will rain the day after tomorrow.

In a class, there are 4 first-year boys, 6 first-year 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?

Suppose that you continually collect coupons and that there are \(m\) different types. Suppose also that each time a new coupon is obtained, it is a type \(i\) coupon with probability \(p_{i}, i=1, \ldots, m .\) Suppose that you have just collected your \(n\)th coupon. What is the probability that it is a new type? Hint: Condition on the type of this coupon.

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 days the stock will be at its original price? (b) What is the probability that after 3 days the stock's price will have increased by 1 unit? (c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up on the first day?

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 that lands on heads with some unknown probability \(p\) that need not be equal to \(\frac{1}{2} .\) Consider the following procedure for accomplishing our task: 1\. Flip the coin. 2\. Flip the coin again. 3\. If both flips land on heads or both land on tails, return to step 1. 4\. Let the result of the last flip be the result of the experiment. (a) Show that the result is equally likely to be either heads or tails. (b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?

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 \(H, H, H, H ?\) Hint for part \((c):\) 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 person will have brown eyes. (Because of the latter fact, we say that the brown-eyed gene is dominant over the blue-eyed one.) A newborn child independently receives one eye gene from each of its parents, and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent. Suppose that Smith and both of his parents have brown eyes, but Smith's sister has blue eyes. (a) What is the probability that Smith possesses a blue eyed gene? (b) Suppose that Smith's wife has blue eyes. What is the probability that their first child will have blue eyes? (c) If their first child has brown eyes, what is the probability that their next child will also have brown eyes?

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 two children, exactly one of whom is an albino. Suppose that the nonalbino child mates with a person who is known to be a carrier for albinism. (a) What is the probability that their first offspring is an albino? (b) What is the conditional probability that their second offspring is an albino given that their firstborn is not?

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits a wooden duck target with probability \(p_{1},\) while each shot of Dianne's hits it with probability \(p_{2} .\) Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that (a) both shots hit the duck? (b) Barbara's shot hit the duck? What independence assumptions have you made?

\(A\) 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 shot of \(A\) will hit \(B\) with probability \(p_{A},\) and each shot of \(B\) will hit \(A\) with probability \(p_{B}\). What is (a) the probability that \(A\) is not hit? (b) the probability that both duelists are hit? (c) the probability that the duel ends after the \(n\) th round of shots? (d) the conditional probability that the duel ends after the \(n\) th round of shots given that \(A\) is not hit? (e) the conditional probability that the duel ends after the nth round of shots given that both duelists are hit?