A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 4, 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 you are to guess which card it will be. After making your guess, you are told whether or not your guess is correct but not which card was turned over. It turns out that the strategy that maximizes the expected number of correct guesses fixes a permutation of the n cards, say 1, 2,. . ., n, and then continually guesses 1 until it is correct, then continually guesses 2 until either it is correct or all cards have been turned over, and then continuality guesses 3, and so on. Let G denote the number of correct guesses yielded by this strategy. Determine
Hint: In order for C to be at least k what must be the order of cards 1,…,k.
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Chapter 4 Solutions
A First Course in Probability
Ch. 4 - Two balls are chosen randomly from an urn...Ch. 4 - Two fair dice are rolled, Let X equal the product...Ch. 4 - Three dice are rolled. By assuming that each of...Ch. 4 - Five men and 5 women are ranked according to their...Ch. 4 - Let X represent the difference between the number...Ch. 4 - In Problem 4.5 for n=3, if the coin is assumed...Ch. 4 - Suppose that a die is rolled twice. What are the...Ch. 4 - If the die in Problem 4.7 is assumed fair,...Ch. 4 - Repeat Example 1c, when the balls are selected...Ch. 4 - Let X be the winnings of a gambler. Let...
Ch. 4 - The random variable X is said to follow the...Ch. 4 - In the game of Two-Finger Morra, 2 players show 1...Ch. 4 - A salesman has scheduled two appointments to sell...Ch. 4 - Five distinct numbers are randomly distributed to...Ch. 4 - The National Basketball Association (NBA) draft...Ch. 4 - A deck of n cards numbered 1 through n are to be...Ch. 4 - Suppose that the distribution function of X is...Ch. 4 - Four independent flips of a fair coin are made....Ch. 4 - If the distribution function of X is given...Ch. 4 - A gambling book recommends the following winning...Ch. 4 - Four buses carrying 148 students from the same...Ch. 4 - Suppose that two teams play a series of games that...Ch. 4 - You have $1000, and a certain commodity presently...Ch. 4 - A and B play the following game: A writes down...Ch. 4 - Prob. 4.25PCh. 4 - One of the numbers I through 10 is randomly...Ch. 4 - An insurance company writes a policy to the effect...Ch. 4 - A sample of 3 items is selected at random from a...Ch. 4 - There are two possible causes for a breakdown of a...Ch. 4 - A person tosses a fair coin until a tail appears...Ch. 4 - 4.31. Each night different meteorologists give us...Ch. 4 - To determine whether they have a certain disease,...Ch. 4 - A newsboy purchases papers at 10 cents and sells...Ch. 4 - Prob. 4.34PCh. 4 - A box contains 5 red and 5 blue marbles. Two...Ch. 4 - Consider Problem 4.22 t with i=2. Find the...Ch. 4 - Find Var (X) and Var (Y) for X and as given in...Ch. 4 - If E[X]=1 and var(X)=5, find a. E[(2+X)2]; b....Ch. 4 - A ball is drawn from an urn containing 3 white and...Ch. 4 - On a multiple-choice exam with 3 possible answers...Ch. 4 - A man claims to have extrasensory perception. As a...Ch. 4 - A and B will take the same 10-question...Ch. 4 - A communications channel transmits the digits 0...Ch. 4 - A satellite system consists of n components and...Ch. 4 - A student is getting ready to take an important...Ch. 4 - Suppose that it takes at least 9 votes from a...Ch. 4 - In some military courts, 9 judges are appointed....Ch. 4 - It is known that diskettes produced by a certain...Ch. 4 - When coin 1 is flipped, it lands on heads with...Ch. 4 - Suppose that a biased coin that lands on heads...Ch. 4 - The expected number of typographical errors on a...Ch. 4 - The monthly worldwide average number of airplane...Ch. 4 - Approximately 80000 marriages took place in the...Ch. 4 - State your assumptions. Suppose that the average...Ch. 4 - A certain typing agency employs 2 typists. The...Ch. 4 - How many people are needed so that the probability...Ch. 4 - Suppose that the number of accidents occurring on...Ch. 4 - Compare the Poisson approximation with the correct...Ch. 4 - If you buy a lottery ticket in 50 lotteries, in...Ch. 4 - The number of times that a person contracts a cold...Ch. 4 - The probability of being dealt a full house in a...Ch. 4 - Consider n, independent trials, each of which...Ch. 4 - People enter a gambling casino at a rate of 1...Ch. 4 - The suicide rate in a certain state is 1 suicide...Ch. 4 - Each of 500 soldiers in an army company...Ch. 4 - A total of 2n people, consisting of n married...Ch. 4 - Prob. 4.67PCh. 4 - In response to an attack of 10 missiles, 500...Ch. 4 - A fair coin is flipped 10 times. Find the...Ch. 4 - At time 0, a coin that comes up heads with...Ch. 4 - Consider a roulette wheel consisting of 38 numbers...Ch. 4 - Two athletic teams play a series of games; the...Ch. 4 - Suppose in Problem 4.75 that the two teams are...Ch. 4 - An interviewer is given a list of people she can...Ch. 4 - Prob. 4.75PCh. 4 - Solve the Banach match problem (Example 8e) when...Ch. 4 - In the Banach matchbox problem, find the...Ch. 4 - An urn contains 4 white and 4 black balls. We...Ch. 4 - Suppose that a batch of 100 items contains 6 that...Ch. 4 - A game popular in Nevada gambling casinos is Keno,...Ch. 4 - In Example 81 what percentage of i defective lots...Ch. 4 - A purchaser of transistors buys them in lots of...Ch. 4 - There are three highways in the county. The number...Ch. 4 - Suppose that 10 balls are put into 5 boxes, with...Ch. 4 - There are k types of coupons. Independently of the...Ch. 4 - There are N distinct types of coupons, and each...Ch. 4 - If X has distribution function F, what is the...Ch. 4 - If X has distribution function F, what is the...Ch. 4 - The random variable X is said to have the...Ch. 4 - Let N be a nonnegative integer-valued random...Ch. 4 - Let X be such that P{X=1}=p=1P{X=1}. Find c1 such...Ch. 4 - Let X be a random variable having expected value ...Ch. 4 - Find Var (X) if P(X=a)=(1)=p=1P(X=b)Ch. 4 - Show how the derivation of the binomial...Ch. 4 - Let X be a binomial random variable with...Ch. 4 - Consider n independent sequential trials, each of...Ch. 4 - There are n components lined up in a linear...Ch. 4 - Let X be a binomial random variable with...Ch. 4 - A family has n children with probability pn,n1...Ch. 4 - Suppose that n independent tosses of a coin having...Ch. 4 - Let X be a Poisson random variable with parameter...Ch. 4 - Let X be a Poisson random variable with parameter ...Ch. 4 - Prob. 4.18TECh. 4 - Show that X is a Poisson random variable with...Ch. 4 - Consider n coins, each of which independently...Ch. 4 - From a set of n randomly chosen people, let Eij...Ch. 4 - An urn contains 2 n balls, of which 2 are numbered...Ch. 4 - Consider a random collection of n individuals. In...Ch. 4 - Here is another way to obtain a set of recursive...Ch. 4 - Suppose that the number of events that occur in a...Ch. 4 - Prove i=0nii!=1n!exxndx Hint: Use integration by...Ch. 4 - If X is a geometric random variable, show...Ch. 4 - Let X be a negative binomial random variable with...Ch. 4 - For a hyper geometric random variable,...Ch. 4 - Balls numbered I through N are in an urn. Suppose...Ch. 4 - A jar contains m+n chips, numbered 1, 2,. ., n+m....Ch. 4 - Prob. 4.32TECh. 4 - Prob. 4.33TECh. 4 - Prob. 4.34TECh. 4 - An urn initially contains one red and one blue...Ch. 4 - Prob. 4.36TECh. 4 - Prob. 4.1STPECh. 4 - Prob. 4.2STPECh. 4 - A coin that when flipped comes up heads with...Ch. 4 - Prob. 4.4STPECh. 4 - Suppose that P{X=0}=1P{X=1}. If E[X]=3Var(X), find...Ch. 4 - There are 2 coins in a bin. When one of them is...Ch. 4 - Prob. 4.7STPECh. 4 - Prob. 4.8STPECh. 4 - Prob. 4.9STPECh. 4 - An urn contains n balls numbered 1 through n. If...Ch. 4 - Prob. 4.11STPECh. 4 - Prob. 4.12STPECh. 4 - Each of the members of a 7-judge panel...Ch. 4 - Prob. 4.14STPECh. 4 - The number of eggs laid on a tree leaf by an...Ch. 4 - Each of n boys and n girls, independently and...Ch. 4 - A total of 2n people, consisting of n married...Ch. 4 - Prob. 4.18STPECh. 4 - Prob. 4.19STPECh. 4 - Show that if X is a geometric random variable with...Ch. 4 - Suppose that P{X=a}=p,P{X=b}=1p a. Show that Xbab...Ch. 4 - Prob. 4.22STPECh. 4 - Balls are randomly withdrawn, one at a time...Ch. 4 - Ten balls are to be distributed among 5 urns, with...Ch. 4 - For the match problem (Example 5m in Chapter 2),...Ch. 4 - Let be the probability that a geometric random...Ch. 4 - Two teams will play a series of games, with the...Ch. 4 - An urn has n white and m black balls. Balls are...
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