Problem 3.1P: Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given... Problem 3.2P: If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given... Problem 3.3P: Use Equation (2.1) to compute in a hand of bridge the conditional probability that East has 3 spades... Problem 3.4P: What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of... Problem 3.5P: An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without... Problem 3.6P: Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with... Problem 3.7P: The king comes from a family of 2 children. What is the probability that the other child is his... Problem 3.8P: A couple has 2 children. What is the probability that both are girls if the older of the two is a... Problem 3.9P: Consider 3 urns. Urn A contains 2 white and 4 red balls, urn B contains 8 white and 4 red balls, and... Problem 3.10P: Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards.... Problem 3.11P: Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let B be the... Problem 3.12P: Suppose distinct values are written on each of 3 cards, which are then randomly given the... Problem 3.13P: A recent college graduate is planning to take the first three actuarial examinations in the coming... Problem 3.14P: Suppose that an ordinary deck of 52 cards (which contains 4 aces is randomly divided into 4 hands of... Problem 3.15P: An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is... Problem 3.16P: An ectopic pregnancy is twice as likely to develop when the pregnant woman is a smoker as it is when... Problem 3.17P: Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve... Problem 3.18P: In a certain community, 36 percent of the families own a dog and 22 percent of the families that own... Problem 3.19P: A total of 46 percent of the voters in a certain city classify themselves as Independents, whereas... Problem 3.20P: A total of 4.8 percent of the women and 37 percent of the men o took a certain quit smoking class... Problem 3.21P: Fifty-two percent of the students at a certain college are females. Five percent of the students in... Problem 3.22P: A total of 500 married working couples were polled about their annual salaries, with the following... Problem 3.23P: A red die, a blue die, and a yellow die (all six sided) are rolled. We are interested in the... Problem 3.24P: Urn I contains 2 white and 4 red balls, whereas urn II contains I white and 1 red ball A ball is... Problem 3.25P: Twenty percent of Bs phone calls are with her daughter. Sixty five percent of the time that B speaks... Problem 3.26P: Each of 2 balls is painted either black or gold and then placed in an urn. Suppose that each ball is... Problem 3.27P: The following method was proposed to estimate the number of people over the age of 50 who reside in... Problem 3.28P: Suppose that 5 percent of men and 0.25 percent of women are color blind. A color-blind person is... Problem 3.29P: All the workers at a certain company drive to work and park in the companys lot. The company is... Problem 3.30P: Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a... Problem 3.31P: There are 15 tennis balls in a box, of which 9 have not previously been used. Three of the balls are... Problem 3.32P: Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble.... Problem 3.33P: Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend... Problem 3.34P: A family has j children with probability pj, where p1=.1,p2=.25,p3=.35,p4=.3. A child from this... Problem 3.35P: On rainy days, Joe is late to work with probability .3: on nonrainy days, he is late with... Problem 3.36P: In Example 31, suppose that the new evidence is subject to different possible interpretations and in... Problem 3.37P: With probability .6, the present was hidden by mom: with probability .4, it was hidden by dad. When... Problem 3.38P: Stores A, B, and C have 50, 75, and 100 employees, respectively, and 50, 60, and 70 percent of them... Problem 3.39P: a. A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at... Problem 3.40P: Urn A has 5 white and 7 black balls. Urn B has 3 white and 12 black balls. We flip a fair coin. If... Problem 3.41P: In Example 3a, what is the probability that someone has an accident in the second year given that he... Problem 3.42P: Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white... Problem 3.43P: A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from... Problem 3.44P: Twelve percent of all U.S. households are In California. A total of 1.3 percent of all U.S.... Problem 3.45P: There are 3 coins in a box. One is a two-headed coin, another is a fair coin, arid the third is a... Problem 3.46P: Three prisoners are informed by their jailer that one of them has been chosen at random to be... Problem 3.47P: There is a 30 percent chance that A can fix her busted computer. If A cannot, then there is a 40... Problem 3.48P: In any given year, a male automobile policyholder will make a claim with probability pm and a female... Problem 3.49P: An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is... Problem 3.50P: Each of 2 cabinets identical n appearance has 2 drawers. Cabinet A contains a silver coin in each... Problem 3.51P: Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male... Problem 3.52P: Suppose that an insurance company classifies people into one of three classes: good risks, average... Problem 3.53P: A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that... Problem 3.54P: Players A, B, C, D are randomly lined up. The first two players in line then play a game: the winner... Problem 3.55P: Players 1,2,3 are playing a tournament. Two of these three players are randomly chosen to play a... Problem 3.56P: Suppose there are two coins, with coin 1 landing heads when flipped with probability .3 and coin 2... Problem 3.57P: In a 7 game series played with two teams, the first team to win a total of 4 games is the winner.... Problem 3.58P: A parallel system functions whenever at least one of its components works. Consider a parallel... Problem 3.59P: If you had to construct a mathematical model for events E and F, as described in parts (a) through... Problem 3.60P: In a class, there are 4 first-year boys, 6 first-year girls, and 6 sophomore boys. How many... Problem 3.61P: Suppose that you continually collect coupons and that there are m different types. Suppose also that... Problem 3.62P: A simplified model for the movement of the price of a stock supposes that on each day the stocks... Problem 3.63P: Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our... Problem 3.64P: Independent flips of a coin that lands on heads v.4th probability p are made. What is the... Problem 3.65P: The color of a persons eyes is determined by a single pair of genes if they are both blue-eyed... Problem 3.66P: Genes relating to albinism are denoted by A and a. Only those people who receive the a gene from... Problem 3.67P: Barbara and Dianne go target shooting Suppose that each of Barbaras shots hits a wooden duck target... Problem 3.68P: A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and... Problem 3.69P: Assume, as in Example 3h, that 64 percent of twins are of the same sex. Given that a newborn set of... Problem 3.70P: The probability of the closing of the ith relay in the circuits shown in Figure 3.5 is given by... Problem 3.71P: An engineering system consisting of n components is said to be a k-out-of-n system (kn) if the... Problem 3.72P: In Problem 3.70a, find the conditional probability that relays 1 and 2 are both closed given that a... Problem 3.73P: A certain organism possesses a pair of each of 5 different genes (which we will designate by the... Problem 3.74P: There is a 50—50 chance that the queen carries the gene for hemophilia. If she is a carrier, then... Problem 3.75P: A town council of 7 members contains a steering committee of size 3. New ideas for legislation go... Problem 3.76P: Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of... Problem 3.77P: A and B alternate rolling a pair of dice, stopping either when A rolls the sum 9 or when B rolls the... Problem 3.78P: In a certain village, it is traditional for the eldest son (or the older son in a two-son family)... Problem 3.79P Problem 3.80P: Consider an unending sequence of independent trials, where each trial is equally likely to result in... Problem 3.81P: A and B play a series of games. Each game is independently won by A with probability p and by B with... Problem 3.82P: In successive rolls of a pair of fair dice, what is the probability of getting 2 sevens before 6... Problem 3.83P: In a certain contest, the players are of equal skill and the probability is 12 that a specified one... Problem 3.84P: An investor owns shares in a stock whose present value is 25. She has decided that she must sell her... Problem 3.85P: A and B flip coins. A starts and continues flipping until a tail occurs, at which point B starts... Problem 3.86P: Die A has 4 red and 2 white faces, whereas die B has 2 red and 4 white faces. A fair coin is flipped... Problem 3.87P: An urn contains 12 balls, of which 4 are white. Three players—A, B, and C successively draw from... Problem 3.88P: Repeat Problem 3.87 when each of the 3 players selects from his own urn. That is, suppose that there... Problem 3.89P: Let S={1,2,...,n} and suppose that A and B are, independently, equally likely to be any of the 2n... Problem 3.90P: Consider an eight team tournament with the format given In Figure 3.6. If the probability that team... Problem 3.91P: Consider Example 2a, but now suppose that when the key is in a certain pocket, there is a 10 percent... Problem 3.92P: In Example 5, what is the conditional probability that the with coin was selected given that the... Problem 3.93P: In Laplace s rule of succession (Example 5e ), are the outcomes of the successive flips independent?... Problem 3.94P: A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty.... Problem 3.95P: Each of n workers is independently qualified to do an incoming job with probability p. If none of... Problem 3.96P: Suppose in the preceding problem that n=2 and that worker i is qualified with probability pi,i=1,2.... Problem 3.97P: Each member of a population of size n is, independently of other members, female with probability p... Problem 3.1TE: Show that if P(A)0, then P(ABA)P(ABAB) Problem 3.2TE Problem 3.3TE: Consider a school community of m families, with ni of them having i children,i=1,...,k,i=1kni=m.... Problem 3.4TE: 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,... Problem 3.5TE: a. Prove that if E and F are mutually exclusive, then P(EEF)=P(E)P(E)+P(F). b. Prove that if Ei,i1... Problem 3.6TE: Prove that if E1,E2,...,En are independent events, then P(E1E2...En)=1i=1n[1P(Ei)]. Problem 3.7TE: a. An urn contains n white and m black balls. The balls are withdrawn one at a time until only those... Problem 3.8TE: Let A, B, and C, be events relating to the experiment of rolling a pair of dice. a. If P(AC)P(BC)... Problem 3.9TE: Consider two independent tosses of a fair coin. Let A be the event that the first toss results in... Problem 3.10TE: Two percent of women age 45 who participate in routine screening have breast cancer. Ninety percent... Problem 3.11TE: In each of n independent tosses of a coin, the coin lands on heads with probability p. How large... Problem 3.12TE: Show that 0ai1,i=1,2,..., then i=1[aij=1i=1(1aj)]+i=1(1ai)=1 Hint: Suppose that an infinite number... Problem 3.13TE: The probability of getting a head on a single toss of a coin is p. Suppose that A starts and... Problem 3.14TE: Suppose that you are gambling against an infinitely rich adversary and at each stage you either win... Problem 3.15TE: Independent trials that result in a success with probability pare successively performed until a... Problem 3.16TE: Independent trials that result in a success with probability p and a failure with probability 1p are... Problem 3.17TE: Suppose that n independent trials are performed, with trial i being a success with probability... Problem 3.18TE: Let Q. denote the probability that no run of 3 consecutive heads appears in n tosses of a fair coin.... Problem 3.19TE: Consider the gamblers ruin problem, with the exception that A and B agree to play no more than n... Problem 3.20TE Problem 3.21TE: The Ballot Problem. In an election, candidate A receives n votes and candidate B receives m votes,... Problem 3.22TE: As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow... Problem 3.23TE: A bag contains a white and b black balls. Balls are chosen from the bag according to the following... Problem 3.24TE: A round-robin tournament of n contestants is a tournament in which each of the (n2) pairs of... Problem 3.25TE: Prove directly thatP(EF)=P(EFG)P(GF)+P(EFGC)P(GCF) Problem 3.26TE: Prove the equivalence of Equations (5.11) and (5.12 ). Problem 3.27TE Problem 3.28TE: Prove or give a counterexample, if E1 and E2 are independent, then they are conditionally... Problem 3.29TE: In Laplaces rule of succession (Example 5e ), show that if the first n flips all result in heads,... Problem 3.30TE: In Laplaces rule of succession (Example 5e), suppose that the first n flips resulted in r heads and... Problem 3.31TE: Suppose that a nonmathematical, but philosophically minded, friend of yours claims that Laplaces... Problem 3.1STPE: In a game of bridge, West has no aces What is the probability of his partners having (a) no aces?... Problem 3.2STPE Problem 3.3STPE: How can 20 balls, 10 white and 10 black, be put into two urns so as to maximize the probability of... Problem 3.4STPE Problem 3.5STPE: An urn has r red and w white balls that are randomly removed one at a time. Let Ri be the event that... Problem 3.6STPE: An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is... Problem 3.7STPE: A friend randomly chooses two cards, without replacement, from an ordinary deck of 52 playing cards.... Problem 3.8STPE: Show that P(HE)P(GE)=P(H)P(G)P(EH)P(EG). Suppose that, before new evidence is observed, the... Problem 3.9STPE: You ask your neighbor to water a sickly plant while you are on vacation. Without water, it will die... Problem 3.10STPE: Six balls are to be randomly chosen from an urn containing 8 red, 10 green, and 12 blue balls. a.... Problem 3.11STPE: A type C battery is in working condition with probability, 7, whereas a type D battery is in working... Problem 3.12STPE Problem 3.13STPE: Balls are randomly removed from an urn that initially contains 20 red and 10 blue balls. a. What is... Problem 3.14STPE: A coin having probability .8 of landing on heads is flipped. A observes the result— either heads... Problem 3.15STPE: In a certain species of rats, black dominates over brown. Suppose that a black rat with two black... Problem 3.16STPE: a. In Problem 3.70b, find the probability that a current flows from A to B, by conditioning on... Problem 3.17STPE: For the k-out-of-n system described in Problem 3.71, assume that each component independently works... Problem 3.18STPE Problem 3.19STPE Problem 3.20STPE: Suppose that there are n possible outcomes of a trial, with outcome i resulting with probability... Problem 3.21STPE: If A flips vand B flips n fair coins, show that the probability that A gets more heads than B is 12... Problem 3.22STPE: Prove or give counterexamples to the following statements: a. If E is independent of F and E is... Problem 3.23STPE: Let A and B be events having positive probability. State whether each of the following statements is... Problem 3.24STPE: Rank the following from most likely to least likely to occur: A. A fair coin lands on heads. B.... Problem 3.25STPE: Two local factories, A and B, produce radios. Each radio produced at factory A is defective with... Problem 3.26STPE: Show that if P(AB)=1, then P(BCAC)=1 Problem 3.27STPE Problem 3.28STPE: A total of 2n cards, of which 2 are aces, are to be randomly divided among two players, with each... Problem 3.29STPE: There are n distinct types of coupons, and each coupon obtained is, independently of prior types... Problem 3.30STPE: Show that for any events E and F,P(EEF)P(EF) Hint: Compute P(EEF) by conditioning on whether F... Problem 3.31STPE: a. If the odds of A is 23, what is the probability that A occurs. b. If the odds of A is 5, what is... Problem 3.32STPE Problem 3.33STPE: If the events E, F, G are independent. show that P(EFGC)=P(E). Problem 3.34STPE: Players 1,2,3, are in a contest. Two of them are randomly chosen to play a game in round one, with... Problem 3.35STPE: If 4 balls are randomly chosen from an urn containing 4 red, 5 white, 6 blue, and 7 green balls,... Problem 3.36STPE: In a 4 player tournament, player 1 plays player 2, player 3 plays player 4, with the winners then... Problem 3.37STPE: In a tournament Involving players 1,..., n, players 1 and 2 play a game, with the loser departing... format_list_bulleted
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