Problem 4.1P: Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose... Problem 4.2P: Two fair dice are rolled, Let X equal the product of the 2 dice. Compute P{X=i} for i=1,...,36. Problem 4.3P: Three dice are rolled. By assuming that each of the 63=216 possible outcomes is equally likely, find... Problem 4.4P: Five men and 5 women are ranked according to their scores on an examination. Assume that no two... Problem 4.5P: Let X represent the difference between the number of heads and the number of tails obtained when a... Problem 4.6P: In Problem 4.5 for n=3, if the coin is assumed fair, what are the probabilities associated with the... Problem 4.7P: Suppose that a die is rolled twice. What are the possible values that the following random variables... Problem 4.8P: If the die in Problem 4.7 is assumed fair, calculate the probabilities associated with the random... Problem 4.9P: Repeat Example 1c, when the balls are selected with replacement. Problem 4.10P: Let X be the winnings of a gambler. Let p(i)=P(X=i) and suppose... Problem 4.11P: The random variable X is said to follow the distribution of Benfords Law if... Problem 4.12P: In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number... Problem 4.13P: A salesman has scheduled two appointments to sell vacuum cleaners. His first appointment will lead... Problem 4.14P: Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players... Problem 4.15P: The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst... Problem 4.16P: A deck of n cards numbered 1 through n are to be turned over one a time. Before each card is shown... Problem 4.17P: Suppose that the distribution function of X is given byF(b)={0b0b40b112+b141b211122b313b a. Find... Problem 4.18P: Four independent flips of a fair coin are made. Let X denote the number of heads obtained. Plot the... Problem 4.19P: If the distribution function of X is given byF(b)={0b0120b1351b2452b39103b3.51b3.5 calculate the... Problem 4.20P: A gambling book recommends the following winning strategy for the game of roulette: Bet $1 on red.... Problem 4.21P: Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry,... Problem 4.22P: Suppose that two teams play a series of games that ends when one of them has won i games. Suppose... Problem 4.23P: You have $1000, and a certain commodity presently sells for $2 per ounce. Suppose that after one... Problem 4.24P: A and B play the following game: A writes down either number 1 or number 2, and B must guess which... Problem 4.25P Problem 4.26P: One of the numbers I through 10 is randomly chosen. You are to try to guess the number chosen by... Problem 4.27P: An insurance company writes a policy to the effect that an amount of money A must be paid if some... Problem 4.28P: A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective.... Problem 4.29P: There are two possible causes for a breakdown of a machine. To check the first possibility would... Problem 4.30P: A person tosses a fair coin until a tail appears for the first time. If the tail appears on the nth... Problem 4.31P: 4.31. Each night different meteorologists give us the probability that it will rain the next day. To... Problem 4.32P: To determine whether they have a certain disease, 100 people are to have their blood tested,... Problem 4.33P: A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to... Problem 4.34P Problem 4.35P: A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same... Problem 4.36P: Consider the friendship network described by Figure 4.5 E[f(X)]. Problem 4.37P: Consider Problem 4.22 t with i=2. Find the variance of the number of games played, and show that... Problem 4.38P: Find Var (X) and Var (Y) for X and as given in Problem 4.21. Problem 4.39P: If E[X]=1 and var(X)=5, find a. E[(2+X)2]; b. var(4+3X). Problem 4.40P: A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is... Problem 4.41P: On a multiple-choice exam with 3 possible answers for each of the 5 questions, what is the... Problem 4.42P: A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man... Problem 4.43P: A and B will take the same 10-question examination. Each question will be answered correctly by A... Problem 4.44P: A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted... Problem 4.45P: A satellite system consists of n components and functions on any given day if at least k of the n... Problem 4.46P: A student is getting ready to take an important oral examination and is concerned about the... Problem 4.47P: Suppose that it takes at least 9 votes from a 12-member jury to convict a defendant. Suppose also... Problem 4.48P: In some military courts, 9 judges are appointed. However, both the prosecution and the defense... Problem 4.49P: It is known that diskettes produced by a certain company will be defective with probability .01,... Problem 4.50P: When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on... Problem 4.51P: Each member of a population of size n is, independently, female with probability p or male with... Problem 4.52P: In a tournament involving players 1,2,3,4, players land 2 play a game, with the loser departing and... Problem 4.53P: Suppose that a biased coin that lands on heads with probability p is flipped 10 times. Given that a... Problem 4.54P: The expected number of typographical errors on a page of a certain magazine is .2. What is the... Problem 4.55P: The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the... Problem 4.56P: Approximately 80000 marriages took place in the state of New York last year. Estimate the... Problem 4.57P: State your assumptions. Suppose that the average number of cars abandoned weekly on a certain... Problem 4.58P: A certain typing agency employs 2 typists. The average number of errors per article is 3 when typed... Problem 4.59P: How many people are needed so that the probability that at least one of them has the same birthday... Problem 4.60P: Suppose that the number of accidents occurring on a highway each day is a Poisson random variable... Problem 4.61P: Compare the Poisson approximation with the correct binomial probability for the following cases: a.... Problem 4.62P: If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is... Problem 4.63P: The number of times that a person contracts a cold in a given year is a Poisson random variable with... Problem 4.64P: The probability of being dealt a full house in a hand of poker is approximately .0014. Find an... Problem 4.65P: Consider n, independent trials, each of which results in one of the outcomes 1.. . ..k with... Problem 4.66P: People enter a gambling casino at a rate of 1 every 2 minutes. a. What is the probability that no... Problem 4.67P: The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. a. Find the... Problem 4.68P: Each of 500 soldiers in an army company independently has a certain disease with probability 1103.... Problem 4.69P: A total of 2n people, consisting of n married couples, are randomly seated (all possible orderings... Problem 4.70P Problem 4.71P: In response to an attack of 10 missiles, 500 antiballistic missiles are launched. The missile... Problem 4.72P: A fair coin is flipped 10 times. Find the probability that there is a string of 4 consecutive heads... Problem 4.73P: At time 0, a coin that comes up heads with probability p is flipped and falls to the ground. Suppose... Problem 4.74P: Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always... Problem 4.75P: Two athletic teams play a series of games; the first team to win 4 games is declared the overall... Problem 4.76P: Suppose in Problem 4.75 that the two teams are evenly matched and each has probability 12 of winning... Problem 4.77P: An interviewer is given a list of people she can interview. If the interviewer needs to interview 5... Problem 4.78P Problem 4.79P: Solve the Banach match problem (Example 8e) when the left-hand matchbox originally contained N1... Problem 4.80P: In the Banach matchbox problem, find the probability that at the moment when the first box is... Problem 4.81P: An urn contains 4 white and 4 black balls. We randomly choose 4 balls, If 2 of them are white and 2... Problem 4.82P: Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X... Problem 4.83P: A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are... Problem 4.84P: In Example 81 what percentage of i defective lots does the purchaser reject? Find it for i = 1. 4.... Problem 4.85P: A purchaser of transistors buys them in lots of 20. It is his policy to randomly inspect 4... Problem 4.86P: There are three highways in the county. The number of daily accidents that occur on these highways... Problem 4.87P: Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box i with... Problem 4.88P: There are k types of coupons. Independently of the types of previously collected coupons, each new... Problem 4.89P: An urn contains 10 red, S black, and 7 green balls. One of the colors is chosen at random (meaning... Problem 4.1TE: There are N distinct types of coupons, and each time one is obtained it will, independently of past... Problem 4.2TE: If X has distribution function F, what is the distribution function of eX? Problem 4.3TE: If X has distribution function F, what is the distribution function of the random variable x+, where... Problem 4.4TE: The random variable X is said to have the Yule-Simons distribution if P{X=n}=4n(n+1)(n+2),n1 a. Show... Problem 4.5TE: Let N be a nonnegative integer-valued random variable. For nonnegative values aj,j1, show that... Problem 4.6TE: Let X be such that P{X=1}=p=1P{X=1}. Find c1 such that E[cx]=1. Problem 4.7TE: Let X be a random variable having expected value and variance 2. Find the expected value and... Problem 4.8TE: Find Var (X) if P(X=a)=(1)=p=1P(X=b) Problem 4.9TE: Show how the derivation of the binomial probabilities P{X=i}=(ni)pi(1p)ni,i=0,...,n leads to a proof... Problem 4.10TE: Let X be a binomial random variable with parameters n and p. Show thatE[1X+1]=1(1p)n+1(n+1)p Problem 4.11TE: Let X be the number of successes that result from 2n independent trials, when each trial is a... Problem 4.12TE: Consider n independent sequential trials, each of which is successful with probability p. If there... Problem 4.13TE: There are n components lined up in a linear arrangement. Suppose that each component independently... Problem 4.14TE: Let X be a binomial random variable with parameters (n,p). What value of p maximizes... Problem 4.15TE: A family has n children with probability pn,n1 where (1p)p. a. What proportion of families has no... Problem 4.16TE: Suppose that n independent tosses of a coin having probability p of coming up heads are made. Show... Problem 4.17TE: Let X be a Poisson random variable with parameter . Show that P{X=i} increases monotonically and... Problem 4.18TE: Let X be a Poisson random variable with parameter a. Show that P{Xiseven}=12[1+e2] by using the... Problem 4.19TE Problem 4.20TE: Show that X is a Poisson random variable with parameter , then E[Xn]=E[(X+1)n1] Now use this result... Problem 4.21TE: Consider n coins, each of which independently comes up heads with probability p. Suppose that n is... Problem 4.22TE: From a set of n randomly chosen people, let Eij denote the event that persons i and j have the same... Problem 4.23TE: An urn contains 2 n balls, of which 2 are numbered 1, 2 are numbered 2, .. ,, and 2 are numbered n.... Problem 4.24TE: Consider a random collection of n individuals. In approximating the probability that no 3 of these... Problem 4.25TE: Here is another way to obtain a set of recursive equations for determining Pn, the probability that... Problem 4.26TE: Suppose that the number of events that occur in a specifiedime is a Poisson random variable with... Problem 4.27TE: Prove i=0nii!=1n!exxndx Hint: Use integration by parts. Problem 4.28TE: If X is a geometric random variable, show analytically that P{X=n+kXn}=P{X=k}. Using the... Problem 4.29TE: Let X be a negative binomial random variable with parameters n and p, and let Y be a binomial random... Problem 4.30TE: For a hyper geometric random variable, determinePX=h+1)P{X=k} Problem 4.31TE: Balls numbered I through N are in an urn. Suppose that n,nN, of them are randomly selected without... Problem 4.32TE: A jar contains m+n chips, numbered 1, 2,. ., n+m. A set of size n, is drawn. If we let X denote the... Problem 4.33TE Problem 4.34TE Problem 4.35TE Problem 4.36TE: An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and... Problem 4.37TE Problem 4.1STPE Problem 4.2STPE Problem 4.3STPE: A coin that when flipped comes up heads with probability p is flipped until either heads or tails... Problem 4.4STPE Problem 4.5STPE: Suppose that P{X=0}=1P{X=1}. If E[X]=3Var(X), find P{X=0}. Problem 4.6STPE: There are 2 coins in a bin. When one of them is flipped, it lands on heads with probability .6, and... Problem 4.7STPE Problem 4.8STPE Problem 4.9STPE Problem 4.10STPE: An urn contains n balls numbered 1 through n. If you withdraw n balls randomly in sequence, each... Problem 4.11STPE Problem 4.12STPE Problem 4.13STPE: Each of the members of a 7-judge panel independently makes a correct decision with probability .7.... Problem 4.14STPE Problem 4.15STPE: The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable... Problem 4.16STPE: Each of n boys and n girls, independently and randomly, chooses a member of the other sex. If a boy... Problem 4.17STPE: A total of 2n people, consisting of n married couples, are randomly divided into n pairs.... Problem 4.18STPE Problem 4.19STPE Problem 4.20STPE: Show that if X is a geometric random variable with parameter p, then E[1X]=plog(p)1p Hint: You will... Problem 4.21STPE: Suppose that P{X=a}=p,P{X=b}=1p a. Show that Xbab is a Bernoulli random variable. b. Find Var(X). Problem 4.22STPE Problem 4.23STPE: Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N... Problem 4.24STPE: Ten balls are to be distributed among 5 urns, with each ball going into urn i with probability... Problem 4.25STPE: For the match problem (Example 5m in Chapter 2), find a. the expected number of matches. b. the... Problem 4.26STPE: Let be the probability that a geometric random variable X with parameter p is an even number. a.... Problem 4.27STPE: Two teams will play a series of games, with the winner being the first team to win a total of 4... Problem 4.28STPE: An urn has n white and m black balls. Balls are randomly withdrawn, without replacement, until a... Problem 4.29STPE Problem 4.30STPE: If X is a binomial random variable with parameters n and p, what type of random variable is nX. Problem 4.31STPE: Let X be the ith smallest number in a random sample of n of the numbers 1.. . .. n+m. Find the... Problem 4.32STPE: Balls are randomly removed from an urn consisting of n red and m blue balls. Let X denote the number... format_list_bulleted