A First Course in Probability

Sheldon M. Ross

Chapter 3

Conditional Probability And Independence - all with Video Answers

Educators

Section 1

Problems

Problem 1

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?

Wendi Zhao

Wendi Zhao

Numerade Educator

Problem 2

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 .

Wendi Zhao

Numerade Educator

Problem 3

Use Equation (2.1) to compute, in a hand of bridge, the conditional probability that East has 3 spades given that North-South have a combined total of 8 spades.

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 4

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 $i, i=2,3, \ldots, 12 ?$

Wendi Zhao

Numerade Educator

Problem 5

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?

Wendi Zhao

Numerade Educator

Problem 6

Consider an urn containing 12 balls of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white, given that the sample drawn contains exactly 3 white balls?

Prashant Bana

Numerade Educator

Problem 7

The king comes from a family of 2 children. What is the probability that the other child is his sister?

Wendi Zhao

Numerade Educator

Problem 8

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

Wendi Zhao

Numerade Educator

Problem 9

Consider 3 urns. Urn $A$ contains 2 white and 4 red balls; urn $B$ contains 8 white and 4 red balls; and um $C$ 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?

Wendi Zhao

Numerade Educator

Problem 10

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.

Wendi Zhao

Numerade Educator

Problem 11

An ectopic pregnancy is twice as likely to develop when the pregnant woman is a smoker as it is when she is a nonsmoker. If 32 percent of women of childbearing age are smokers, what percentage of women having ectopic pregnancies are smokers?

Wendi Zhao

Numerade Educator

Problem 12

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 $\mathrm{C}$ section, what is the probability that her baby survives?

Wendi Zhao

Numerade Educator

Problem 13

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?

Wendi Zhao

Numerade Educator

Problem 14

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 as 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 fraction of voters participated in the local election?

Wendi Zhao

Numerade Educator

Problem 15

A total of 48 percent of the women and 37 percent of the men that took a certain "quit smoking" class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class were male,
(a) what percentage of those attending the party were women?
(b) what percentage of the original class attended the party?

Wendi Zhao

Numerade Educator

Problem 16

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) this 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.

Wendi Zhao

Numerade Educator

Problem 17

A total of 500 married working couples were polled about their annual salaries, with the following information resulting.
Thus, for instance, in 36 of the couples the wife earned more and the husband earned less than $\$ 25,000$. If one of the couples is randomly chosen, what is (a) the probability that the husband earns less than $\$ 25,000$;
(b) the conditional probability that the wife earns more than $\$ 25,000$ given that the husband earns more than this amount;
(c) the conditional - probability that the wife earns more than $\$ 25,000$ given that the husband earns less than this amount?

Wendi Zhao

Numerade Educator

Problem 18

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 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.

Wendi Zhao

Numerade Educator

Problem 19

Suppose that an ordinary deck of 52 cards (which contains 4 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.

Wendi Zhao

Numerade Educator

Problem 20

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 white;
(b) of the first 4 balls selected, exactly 2 are black.

Narayan Hari

Numerade Educator

Problem 21

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 um $\Pi$, 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?

Wendi Zhao

Numerade Educator

Problem 22

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 um tips over and 1 ball falls out. It is painted gold. What is the probability that both balls are gold in this case? Explain.

Wendi Zhao

Numerade Educator

Problem 23

The following method was proposed to estimate the number of people over the age of 50 that reside in a town of known population 100,000 . "As you walk along the streets, keep a running count of the percentage of people that you encounter who are over $50 .$ Do this for a few days; then multiply the obtained percentage by 100,000 to obtain the estimate." Comment on this method.
HINT: Let $p$ denote the proportion of people in this 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 it approximately equal to $p ?$

Shu Naito

Numerade Educator

Problem 24

Suppose that 5 percent of men and $.25$ percent of women are color blind. A colorblind person is chosen at random. What is the probability of this person being mal?? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

Wendi Zhao

Numerade Educator

Problem 25

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.

Jon Southam

Numerade Educator

Problem 26

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 20 th card to appear, what is the conditional probability that the card following it is the
(a) ace of spades;
(b) two of clubs?

Narayan Hari

Numerade Educator

Problem 27

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.

Wendi Zhao

Numerade Educator

Problem 28

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 at random from the selected box. 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?

Wendi Zhao

Numerade Educator

Problem 29

English and American spellings are rigour and rigor, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are Americans, what is the probability that the writer is an Englishman?

Wendi Zhao

Numerade Educator

Problem 30

In Example 3e, suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is 90 percent likely that the criminal possesses this certain characteristic. In this case how likely would it be that the suspect is guilty (assuming, as before, that he has this characteristic)?

Sheryl Ezze

Numerade Educator

Problem 31

One probability class of 30 students contains 15 that are good, 10 that are average, and 5 that are of poor quality. A second probability class, also of 30 students, contains 5 that are good, 10 that are fair, and 15 that are poor. You (the expert) are aware of these numbers, but you have no idea which class is which. If you examine one student selected at random from each class and find that the student from class $A$ is a fair student whereas the student from class $B$ is a poor student, what is the probability that class $A$ is the superior class?

Wendi Zhao

Numerade Educator

Problem 32

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

Wendi Zhao

Numerade Educator

Problem 33

(a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?
(b) Suppose that he flips the same coin a second time and again it shows heads. What is now the probability that it is the fair coin?
(c) Suppose that he flips the same coin a third time and it shows tails. What is now the probability that it is the fair coin?

Wendi Zhao

Numerade Educator

Problem 34

Um $A$ has 5 white and 7 black balls. $\operatorname{Urn} B$ 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?

Wendi Zhao

Numerade Educator

Problem 35

In Example $3 \mathrm{a}$, what is the probability that someone has an accident in the second year, given that he or she has had no accidents in the first year?

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 36

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 um 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.

Wendi Zhao

Numerade Educator

Problem 37

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.

Wendi Zhao

Numerade Educator

Problem 38

Three cooks, $A, B$, and $C$, bake a special kind of cake, and with respective probabilities $.02, .03$, and $.05$ it fails to rise. In the restaurant where they work, $A$ bakes 50 percent of these cakes, $B 30$ percent, and $C 20$ percent. What proportion of "failures" is caused by $A ?$

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 39

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?

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 40

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 this 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?

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 41

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?

Philomena Marfo

Philomena Marfo

Numerade Educator

Problem 42

Consider the following game. A deck of cards is shuffled and its cards are turned face up one at a time. At any time you can elect to say "next," and if the next card is the ace of spades, then you win, and if not, then you lose. Of course, if the ace of spades appears before you say "next," then you lose. Also, if there is only one card remaining, the ace of spades hasn't yet appeared, and you have never said "next," then you are a winner (since you will say "next"). Argue that no matter what strategy you employ for deciding when to say "next," your probability of winning is $\frac{1}{52}$.

Xiaomeng Zhang

Numerade Educator

Problem 43

An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is randomly chosen from the um. 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?

Wendi Zhao

Numerade Educator

Problem 44

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?

James Kiss

Numerade Educator

Problem 45

Suppose that there was a cancer diagnostic test that was 95 percent accurate both on those that do and those that do not have the disease. If $.4$ percent of the population have cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so.

Sherrie Fenner

Numerade Educator

Problem 46

Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. Their 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 are good risks, 50 percent are average risks, and 30 percent are bad risks, what proportion of people have accidents in a fixed year? If policyholder $A$ had no accidents in 1997 , what is the probability that he or she is a good (average) risk?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 47

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, or weak are $.7, .2$, and .1, respectively.
(a) How certain is she that she will receive the new job offer?
(6) 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?

Nick Johnson

Numerade Educator

Problem 48

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; given that she is accepted and that she is rejected, of receiving notification on each day of next week are as follows:
She estimates that her probability of being accepted is $.6 .$
(a) What is the probability that mail is received on Monday?
(b) What is the conditional probability that mail is received on Tuesday given that it is not received on Monday?
(c) If there is no mail through Wednesday, what is the conditional probability that she will be accepted?
(d) What is the conditional probability that she will be accepted if mail comes on Thursday?
(e) What is the conditional probability that she will be accepted if no mail arrives that week?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 49

A parallel system functions whenever at least one of its components works. Consider a parallel system of $n$ components and suppose that each component independently works with probability $\frac{1}{2}$. Find the conditional probability that component 1 works given that the system is functioning.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 50

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 over 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.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 51

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?

Hossam Mohamed

Numerade Educator

Problem 52

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?

Hoan Nguyen

Numerade Educator

Problem 53

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 it 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?

Sheryl Ezze

Numerade Educator

Problem 54

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 $\frac{1}{2}$. Consider the following procedure for accomplishing our task.
1. Flip the coin.
2. Flip the coin again.
3. If both fli?s land heads or both land tails, retum 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?

Victor Salazar

Numerade Educator

Problem 55

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 $H, H, H, H$ ?

Ahmad Reda

Numerade Educator

Problem 56

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? Suppose that Smith's wife has blue eyes.
(b) 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?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 57

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?

Bryan Valdivia

Numerade Educator

Problem 57

On the morning of September 31,1982 , the won-lost records of the three leading baseball teams in the western division of the National League of the United States were as follows: Each team had 3 games remaining to be played. All 3 of the Giants games were with the Dodgers, and the 3 remaining games of the Braves were against the San Diego Padres. Suppose that the outcomes of all remaining games are independent and each game is equally likely to be won by either participant. What are the probabilities that each of the teams wins the division? If two teams tie for first place, they have a playoff game, which each team has an equal chance of winning.

Joe Lesueur

Numerade Educator

Problem 58

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits the 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?

Michelle Z.

Numerade Educator

Problem 59

$A$ 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 $n$th round of shots given that both duelists are hit?

Victor Salazar

Numerade Educator

Problem 60

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 this couple?
(a) Choose one of them and let that person answer the question; or
(b) have them both consider the question and then either give the common answer if they agree or, if they disagree, flip a coin to determine which answer to give?

Ahmad Reda

Numerade Educator

Problem 61

In Problem 60 , if $p=.6$ and the couple uses the strategy in part (b), what is the conditional probability that the couple gives the correct answer given that they (a) agree; (b) disagree?

James Kiss

Numerade Educator

Problem 62

The probability of the closing of the $i$ th relay in the circuits shown is given by $p_{i}, i=1,2,3,4,5 .$ If all relays function independently, what is the probability that a current flows between $A$ and $B$ for the respective circuits?

Ahmad Reda

Numerade Educator

Problem 63

An engineering system consisting of $n$ components is said to be a $k$-out-of$n$ system $(k \leq n)$ if the system functions if and only if at least $k$ of the $n$ components function. Suppose that all components function independently of each other.
(a) If the $i$ th component functions with probability $P_{t}, i=1,2,3,4$, compute the probability that a 2-out-of-4 system functions.
(b) Repeat part (a) for a 3-out-of-5 system.
(c) Repeat for a $k$-out-of- $n$ system when all the $P_{i}$ equal $p$ (that is, $P_{i}=p$, $i=1,2, \ldots, n)$

Robin Corrigan

Numerade Educator

Problem 64

In Problem $62 \mathrm{a}$, find the conditional probability that relays 1 and 2 are both closed given that a current flows from $A$ to $B$.

Hunza Gilgit

Numerade Educator

Problem 65

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 that if an organism possesses the gene pair $x X$, then it will outwardly have the appearance of the $X$ gene. For instance, if $X$ stands for brown eyes and $x$ for blue eyes, then an individual having either gene pair $X X$ or $x X$ will have brown eyes, whereas one having gene pair $x x$ will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus 2 organisms with respective genotypes $a A, b B, c c$, $d D, e e$ and $A A, B B, c c, D D, e e$ would have different genotypes but the same phenotype.) In a mating between 2 organisms each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of its mate. In a mating between organisms having genotypes $a A, b B, c C, d D, e E$ and $a a, b B, c c, D d, e e$ what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble
(a) the first parent;
(b) the second parent;
(c) either parent;
(d) neither parent?

Victor Salazar

Numerade Educator

Problem 66

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 the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator

Problem 68

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 vote (of at least 4 ) to pass. Consider now a new piece of legislation and suppose that each town council member will approve it, independently, with probability $p$. What is the probability that a given steering committee member's vote is decisive in the sense that if that person's vote were reversed, then the final fate of the legislation would be reversed? What is the corresponding probability for a given council member not on the steering committee?

Julian Wong

Numerade Educator

Problem 69

Suppose that each child born to a couple is equally likely to be a boy or a girl independent 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 boys and the others girls.
(c) Exactly 3 are boys.
(d) The 2 oldest are girls.
(e) There is at least 1 girl

TH

Timothy Hollman

Numerade Educator

Problem 70

The probability of winning on a single toss of some dice is $p$. A starts, and if he fails, he passes the dice to $B$, who then attempts to win on his roll. They continue to pass the dice back and forth until one of them wins. What are their respective probabilities of winning? Repeat if there are $k$ players.

Nick Johnson

Numerade Educator

Problem 71

Repeat Problem 70 under the assumption that when $A$ rolls the dice, she wins with probability $P_{1}$, and, when $B$ rolls, $B$ wins with probability $P_{2}$.

Aman Gupta

Numerade Educator

Problem 72

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 $P(E) /[P(E)+P(F)]$.

Amany Waheeb

Numerade Educator

Problem 73

When $A$ and $B$ flip coins, the one coming closest to a given line wins 1 penny from the other. If $A$ starts with 3 and $B$ with 7 pennies, what is the probability that $A$ winds up with all of the money if both players are equally skilled? What if $A$ were a better player who won 60 percent of the time?

Hoan Nguyen

Numerade Educator

Problem 74

In successive rolls of a pair of fair dice, what is the probability of getting 2 sevens before 6 even numbers?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 75

Players are of equal skill, and in a contest the probability is $\frac{1}{2}$ that a specified one of the two contestants will be the victor. A group of $2^{n}$ players are paired off against each other at random. The $2^{n-1}$ winners are again paired off randomly, and so on, until a single winner remains. Consider two specified contestants, $A$ and $B$, and define the events $A_{i}, i \leq n, E$ by
$A_{i}: A$ plays in exactly $i$ contests;
$E: A$ and $B$ ever play each other.
(a) Find $P\left(A_{i}\right), i=1, \ldots, n$.
(b) Find $P(E)$.
(c) Let $P_{n}=P(E)$. Show that
$$
P_{n}=\frac{1}{2^{n}-1}+\frac{2^{n}-2}{2^{n}-1}\left(\frac{1}{2}\right)^{2} P_{n-1}
$$
and use this to check your answer obtained in part (b).
For another approach for solving this problem, note that there are a total of $2^{n}-1$ games played.
(d) Explain why a total of $2^{n}-1$ games are played.
Number these games and let $B_{i}$ denote the event that $A$ and $B$ play each other in game $i, i=1, \ldots, 2^{n}-1$.
(e) What is $P\left(B_{i}\right)$ ?
(f) Use nart (e) to find $P(E)$.

Victor Salazar

Numerade Educator

Problem 76

A stock market investor owns shares in a stock whose present value is 25 . She has decided that she must sell her stock if it either goes down to 10 or up to 40 . If each change of price is either up 1 point with probability $.55$ or down 1 point with probability $.45$, and the successive changes are independent, what is the probability that the investor retires a winner?

NM

Nosheen Malik

Numerade Educator

Problem 77

$A$ and $B$ flip coins. A starts and continues flipping until a tail occurs. At this point $B$ starts flipping and continues until there is a tail, then $A$ takes over, and so on. Let $P_{1}$ be the probability of the coin's landing heads when $A$ flips, and $P_{2}$ when $B$ flips. The winner of the game is the first one to get
(a) 2 heads in a row;
(b) a total of 2 heads;
(c) 3 heads in a row;
(d) a total of 3 heads.
In each case, find the probability that $A$ wins.

Victor Salazar

Numerade Educator

Problem 78

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 tails, then die $B$ is to be used.
(a) Show that the probability of red at any throw is $\frac{1}{2}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 79

There are 12 balls, of which 4 are white, in an urn. Three players $-A, B$, $C$-successively draw from the urn, $A$ first, then $B$, then $C$, then $A$, and so on. The winner is the first one to draw a white ball. Find the win probabilities for each player if
(a) each ball is replaced after it is drawn;
(b) the withdrawn balls are not replaced.

Narayan Hari

Numerade Educator

Problem 80

Repeat Problem 79 so that each of the 3 players selects from his own urn. That is, suppose that there are 3 different urns of 12 balls with 4 white in each.

Prashant Bana

Numerade Educator

Problem 81

Let $S=\{1,2, \ldots, n\}$ and suppose that $A$ and $B$ are, independently, equally likely to be any of the $2^{n}$ subsets (including the null set and $S$ itself) of $S$.
(a) Show that
$$
P\{A \subset B\}=\left(\frac{3}{4}\right)^{n}
$$
HINT: Let $N(B)$ denote the number of elements in $B$. Use
$$
P\{A \subset B\}=\sum_{t=0}^{n} P\{A \subset B \mid N(B)=i\} P\{N(B)=i\}
$$
(b) Show that $P\{A B=\varnothing\}=\left(\frac{3}{4}\right)^{n}$.

JW

Julian Wong

Numerade Educator

Problem 82

In Example 5d, what is the conditional probability that the ith coin was selected given that the first $n$ trials all result in heads?

Dalia Rodriguez

Dalia Rodriguez

Numerade Educator

Problem 83

In Laplace's rule of succession, Example $5 \mathrm{~d}$, are the outcomes of the successive flips independent? Explain.

Amany Waheeb

Numerade Educator

Problem 84

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 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that
(a) judges 1 and 2 vote guilty;
(b) judges 1 and 2 cast 1 guilty and 1 not guilty vote;
(c) judges 1 and 2 both cast not guilty votes.
Let $E_{i}, i=1,2,3$ denote the event that judge $i$ casts a guilty vote. Are these events independent. Are they conditionally independent? Explain.

Ahmad Reda

Numerade Educator

Problem 85

Suppose that $n$ independent trials, each of which results in any of the outcomes 0,1 , or 2 with respective probabilities, $p_{0}, p_{1}$, and $p_{2}, \sum_{i=0}^{2} p_{i}=1$, are performed. Find the probability that outcomes 1 and 2 both occur at least once.

Hunza Gilgit

Numerade Educator