A First Course in Probability
A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear re…
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Chapter 7, Problem 7.24P

A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

a. Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

Hint: Define n + m , indicator variables, one for each of the small pills initially present and one for each of the m, small pills created when a large one is split in two. Now use the argument of Example 2m.

b. Let Y denote the day on which the last large pill is chosen. Find E[Y].

Hint: What is the relationship between X and Y?

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Chapter 7 Solutions

A First Course in Probability

Ch. 7 - A player throws a fair die and simultaneously...Ch. 7 - The game of Clue involves 6 suspects, 6 weapons,...Ch. 7 - Gambles are independent, and each one results in...Ch. 7 - Prob. 7.4PCh. 7 - The county hospital is located at the center of a...Ch. 7 - A fair die is rolled 10 times. Calculate the...Ch. 7 - Suppose that A and B each randomly and...Ch. 7 - N people arrive separately to a professional...Ch. 7 - A total of n. balls, numbered 1 through n, are put...Ch. 7 - Consider 3 trials, each having the same...
Ch. 7 - Consider n independent flips of a coin having...Ch. 7 - A group of n men and n women is lined up at...Ch. 7 - A set of 1000 cards numbered 1 through 1000 is...Ch. 7 - An urn has m black balls. At each stage, a black...Ch. 7 - In Example 2h, say that i and j, ij form a matched...Ch. 7 - Let Z be a standard normal random variable, and,...Ch. 7 - A deck of n cards numbered 1 through n is...Ch. 7 - Cards from an ordinary deck of 52 playing cards...Ch. 7 - Prob. 7.19PCh. 7 - Prob. 7.20PCh. 7 - For a group of 100 people, compute a. the expected...Ch. 7 - How many times would you expect to roll a fair die...Ch. 7 - Urn I contains 5 white and 6 black balls, while...Ch. 7 - A bottle initially contains m large pills and n...Ch. 7 - Let X1,X2... be a sequence of independent and...Ch. 7 - If X1,X2,....Xn are independent and identically...Ch. 7 - If 101 items are distributed among 10 boxes, then...Ch. 7 - Prob. 7.28PCh. 7 - There are 4 different types of coupons, the first...Ch. 7 - If X and Y are independent and identically...Ch. 7 - Prob. 7.31PCh. 7 - Prob. 7.32PCh. 7 - If E[X]=1 and Var(X)=5, find a. E[(2+X)2]: b....Ch. 7 - If 10 married couples are randomly seated at a...Ch. 7 - Cards from an ordinary deck are turned face up one...Ch. 7 - Let X be the number of ls and F the number of 2s...Ch. 7 - A die is rolled twice. Let X equal the sum of the...Ch. 7 - Prob. 7.38PCh. 7 - Let X1,... be independent with common mean and...Ch. 7 - Prob. 7.40PCh. 7 - A pond contains 100 fish, of which 30 are carp. If...Ch. 7 - A group of 20 people consisting of 10 men and 10...Ch. 7 - Let X1,X2,...,Xn be independent random variables...Ch. 7 - Between two distinct methods for manufacturing...Ch. 7 - Prob. 7.45PCh. 7 - Consider the following dice game. as played at a...Ch. 7 - Prob. 7.47PCh. 7 - A fair die is successively rolled. Let X and Y...Ch. 7 - There are two misshapen coins in a box; their...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - A population is made up of r disjoint subgroups....Ch. 7 - A prisoner is trapped in a cell containing 3...Ch. 7 - Consider the following dice game: A pair of dice...Ch. 7 - Ten hunters are waiting for ducks to fly by. When...Ch. 7 - The number of people who enter an elevator on the...Ch. 7 - Suppose that the expected number of accidents per...Ch. 7 - A coin having probability p of coming up heads is...Ch. 7 - There are n+1 participants in a game. Each person...Ch. 7 - Each of m+2 players pays 1 unit to a kitty in...Ch. 7 - Prob. 7.61PCh. 7 - Prob. 7.62PCh. 7 - Prob. 7.63PCh. 7 - Type i light bulbs function for a random amount of...Ch. 7 - The number of winter storms in a good year is a...Ch. 7 - In Example 5c, compute the variance of the length...Ch. 7 - Prob. 7.67PCh. 7 - The number of accidents that a person has in a...Ch. 7 - Repeat Problem 7.73 when the proportion of the...Ch. 7 - Consider an urn containing a large number of...Ch. 7 - In problem ,suppose that the coin is tossed n...Ch. 7 - Suppose that in Problem 7.75, we continue to flip...Ch. 7 - In Example 6b, let S denote the signal sent and R...Ch. 7 - In Example 6c y)2].Ch. 7 - The moment generating function of X is given by...Ch. 7 - Let X be the value of the first die and Y the sum...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - Prob. 7.78PCh. 7 - Successive weekly sales, in units of $1,000, have...Ch. 7 - Show that E[(Xa)2] is minimized at a=E[X].Ch. 7 - Suppose that X is a continuous random variable...Ch. 7 - Prob. 7.3TECh. 7 - Let X be a random variable having finite...Ch. 7 - Prob. 7.5TECh. 7 - Prob. 7.6TECh. 7 - We say that X is stochastically larger than Y,...Ch. 7 - Prob. 7.8TECh. 7 - A coin having probability p of landing on heads is...Ch. 7 - Let X1,X2,....Xn be independent and identically...Ch. 7 - Prob. 7.11TECh. 7 - Let X1,X2,... be a sequence of independent random...Ch. 7 - Prob. 7.13TECh. 7 - Prob. 7.14TECh. 7 - Prob. 7.15TECh. 7 - Prob. 7.16TECh. 7 - Prob. 7.17TECh. 7 - In Example 41 t, we showed that the covariance of...Ch. 7 - Show that X and Y are identically distributed and...Ch. 7 - Prob. 7.20TECh. 7 - Prob. 7.21TECh. 7 - Prob. 7.22TECh. 7 - Show that Z is a standard normal random variable...Ch. 7 - Prove the Cauchy-Schwarz inequality, namely,...Ch. 7 - Show that if X and Y are independent, then...Ch. 7 - Prove that E[g(X)YX]=g(X)E[YX].Ch. 7 - Prove that if E[YX=x]=E[Y] for all x, then X and Y...Ch. 7 - Prob. 7.28TECh. 7 - Let X1,...,Xn be independent and identically...Ch. 7 - Consider Example 4f, which is concerned with the...Ch. 7 - An urn initially contains b black and w white...Ch. 7 - For an event A, let IA equal 1 if A occurs and let...Ch. 7 - A coin that lands on heads with probability p is...Ch. 7 - For another approach to Theoretical Exercise 7.34,...Ch. 7 - The probability generating function of the...Ch. 7 - One ball at a time is randomly selected from an...Ch. 7 - Prob. 7.37TECh. 7 - Prob. 7.38TECh. 7 - The best quadratic predictor of Y with respect to...Ch. 7 - Use the conditional variance formula to determine...Ch. 7 - Let X be a normal random variable with parameters...Ch. 7 - It follows from Proposition 6.1 and the fact that...Ch. 7 - Show that for random variables X and Z,...Ch. 7 - Prob. 7.44TECh. 7 - Verify the formula for the moment generating...Ch. 7 - For a standard normal random variable Z, let...Ch. 7 - Prob. 7.47TECh. 7 - Prob. 7.48TECh. 7 - The positive random variable X is said to be a...Ch. 7 - Let X have moment generating function M(t), and...Ch. 7 - Use Table 7.2 to determine the distribution of...Ch. 7 - Show how to compute cov(X,Y) from the joint moment...Ch. 7 - Suppose that X1,...,Xn have a multivariate normal...Ch. 7 - If Z is a standard normal random variable, what is...Ch. 7 - Suppose that Y is a normal random variable with...Ch. 7 - Consider a list of m names, where the same name...Ch. 7 - Prob. 7.2STPECh. 7 - Prob. 7.3STPECh. 7 - Prob. 7.4STPECh. 7 - Prob. 7.5STPECh. 7 - Prob. 7.6STPECh. 7 - Prob. 7.7STPECh. 7 - Prob. 7.8STPECh. 7 - Prob. 7.9STPECh. 7 - Prob. 7.10STPECh. 7 - Prob. 7.11STPECh. 7 - Prob. 7.12STPECh. 7 - Prob. 7.13STPECh. 7 - Prob. 7.14STPECh. 7 - Prob. 7.15STPECh. 7 - Prob. 7.16STPECh. 7 - Prob. 7.17STPECh. 7 - Prob. 7.18STPECh. 7 - There are n items in a box labeled H and m in a...Ch. 7 - Let X be a nonnegative random variable having...Ch. 7 - Let a1,...,an, not all equal to 0, be such that...Ch. 7 - Prob. 7.22STPECh. 7 - Prob. 7.23STPECh. 7 - Prob. 7.24STPECh. 7 - Prob. 7.25STPECh. 7 - Prob. 7.26STPECh. 7 - Prob. 7.27STPECh. 7 - Prob. 7.28STPECh. 7 - Prob. 7.29STPECh. 7 - Prob. 7.30STPECh. 7 - Prob. 7.31STPECh. 7 - Starting with etX=1+tX+t2X22!+t3X33!+...+tnXnn!...
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