9th Edition

ISBN: 9780321794772

Author: Sheldon Ross

Publisher: PEARSON

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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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Textbook Question

Chapter 3, Problem 3.58P

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 that lands on heads with some unknown probability p that need not be equal to 1 2 . Consider the following procedure for accomplishing our task:

1. Flip the coin.

2. Flip the coin again.

3. If both flips land on heads or both land on tails, return 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?

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Chapter 3, Problem 3.57P

Chapter 3, Problem 3.59P

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Chapter 3 Solutions

A First Course in Probability

Ch. 3 - Two fair dice are rolled. What is the conditional...Ch. 3 - If two fair dice are rolled, what is the...Ch. 3 - Use Equation (2.1) to compute in a hand of bridge...Ch. 3 - What is the probability that at least one of a...Ch. 3 - An urn contains 6 white and 9 black balls. If 4...Ch. 3 - Consider an urn containing 12 balls, of which 8...Ch. 3 - The king comes from a family of 2 children. What...Ch. 3 - A couple has 2 children. What is the probability...Ch. 3 - Consider 3 urns. Urn A contains 2 white and 4 red...Ch. 3 - Three cards are randomly selected, without...

Ch. 3 - Two cards are randomly chosen without replacement...Ch. 3 - A recent college graduate is planning to take the...Ch. 3 - Suppose that an ordinary deck of 52 cards (which...Ch. 3 - An urn initially contains 5 white and 7 black...Ch. 3 - An ectopic pregnancy is twice as likely to develop...Ch. 3 - Ninety-eight percent of all babies survive...Ch. 3 - In a certain community, 36 percent of the families...Ch. 3 - A total of 46 percent of the voters in a certain...Ch. 3 - A total of 4.8 percent of the women and 37 percent...Ch. 3 - Fifty-two percent of the students at a certain...Ch. 3 - A total of 500 married working couples were polled...Ch. 3 - A red die, a blue die, and a yellow die (all six...Ch. 3 - Urn I contains 2 white and 4 red balls, whereas...Ch. 3 - Each of 2 balls is painted either black or gold...Ch. 3 - The following method was proposed to estimate the...Ch. 3 - Suppose that 5 percent of men and 0.25 percent of...Ch. 3 - All the workers at a certain company drive to work...Ch. 3 - Suppose that an ordinary deck of 52 cards is...Ch. 3 - There are 15 tennis balls in a box, of which 9...Ch. 3 - Consider two boxes, one containing 1 black and 1...Ch. 3 - Ms. Aquina has just had a biopsy on a possibly...Ch. 3 - A family has j children with probability pj, where...Ch. 3 - On rainy days, Joe is late to work with...Ch. 3 - In Example 31, suppose that the new evidence is...Ch. 3 - With probability .6, the present was hidden by...Ch. 3 - Stores A, B, and C have 50, 75, and 100 employees,...Ch. 3 - a. A gambler has a fair coin and a two-headed coin...Ch. 3 - Urn A has 5 white and 7 black balls. Urn B has 3...Ch. 3 - In Example 3a, what is the probability that...Ch. 3 - Consider a sample of size 3 drawn in the following...Ch. 3 - A deck of cards is shuffled and then divided into...Ch. 3 - Twelve percent of all U.S. households are In...Ch. 3 - There are 3 coins in a box. One is a two-headed...Ch. 3 - Three prisoners are informed by their jailer that...Ch. 3 - Suppose we have 10 coins such that if the ith coin...Ch. 3 - Prob. 3.46P Ch. 3 - Prob. 3.47P Ch. 3 - Each of 2 cabinets identical n appearance has 2...Ch. 3 - Prostate cancer is the most common type of cancer...Ch. 3 - Suppose that an insurance company classifies...Ch. 3 - A worker has asked her supervisor for a letter of...Ch. 3 - A high school student is anxiously waiting to...Ch. 3 - A parallel system functions whenever at least one...Ch. 3 - If you had to construct a mathematical model for...Ch. 3 - In a class, there are 4 first-year boys, 6...Ch. 3 - Suppose that you continually collect coupons and...Ch. 3 - A simplified model for the movement of the price...Ch. 3 - Suppose that we want to generate the outcome of...Ch. 3 - Independent flips of a coin that lands on heads...Ch. 3 - The color of a persons eyes is determined by a...Ch. 3 - Genes relating to albinism are denoted by A and a....Ch. 3 - Barbara and Dianne go target shooting Suppose that...Ch. 3 - A and B are involved in a duel. The rules of the...Ch. 3 - A true - false question is to be posed to a...Ch. 3 - Assume, as in Example 3h, that 64 percent of twins...Ch. 3 - The probability of the closing of the ith relay in...Ch. 3 - An engineering system consisting of n components...Ch. 3 - In Problem 3.70a, find the conditional probability...Ch. 3 - A certain organism possesses a pair of each of 5...Ch. 3 - There is a 50â€”50 chance that the queen carries...Ch. 3 - On the morning of September 30, 1982, the...Ch. 3 - A town council of 7 members contains a steering...Ch. 3 - Suppose that each child born to a couple is...Ch. 3 - A and B alternate rolling a pair of dice, stopping...Ch. 3 - In a certain village, it is traditional for the...Ch. 3 - Prob. 3.76P Ch. 3 - Consider an unending sequence of independent...Ch. 3 - A and B play a series of games. Each game is...Ch. 3 - In successive rolls of a pair of fair dice, what...Ch. 3 - In a certain contest, the players are of equal...Ch. 3 - An investor owns shares in a stock whose present...Ch. 3 - A and B flip coins. A starts and continues...Ch. 3 - Die A has 4 red and 2 white faces, whereas die B...Ch. 3 - An urn contains 12 balls, of which 4 are white....Ch. 3 - Repeat Problem 3.87 when each of the 3 players...Ch. 3 - Let S={1,2,...,n} and suppose that A and B are,...Ch. 3 - Consider Example 2a, but now suppose that when the...Ch. 3 - In Example 5, what is the conditional probability...Ch. 3 - In Laplace s rule of succession (Example 5e ), are...Ch. 3 - A person tried by a 3-judge panel is declared...Ch. 3 - Suppose that n independent trials, each of which...Ch. 3 - Show that if P(A)0, then P(ABA)P(ABAB)Ch. 3 - Prob. 3.2TE Ch. 3 - Consider a school community of m families, with ni...Ch. 3 - A ball is in any one of n boxes and is in the ith...Ch. 3 - a. Prove that if E and F are mutually exclusive,...Ch. 3 - Prove that if E1,E2,...,En are independent events,...Ch. 3 - a. An urn contains n white and m black balls. The...Ch. 3 - Let A, B, and C, be events relating to the...Ch. 3 - Consider two independent tosses of a fair coin....Ch. 3 - Two percent of women age 45 who participate in...Ch. 3 - In each of n independent tosses of a coin, the...Ch. 3 - Show that 0ai1,i=1,2,..., then...Ch. 3 - The probability of getting a head on a single toss...Ch. 3 - Suppose that you are gambling against an...Ch. 3 - Independent trials that result in a success with...Ch. 3 - Prob. 3.16TE Ch. 3 - Prob. 3.17TE Ch. 3 - Let Q. denote the probability that no run of 3...Ch. 3 - Consider the gamblers ruin problem, with the...Ch. 3 - Prob. 3.20TE Ch. 3 - The Ballot Problem. In an election, candidate A...Ch. 3 - As a simplified model for weather forecasting,...Ch. 3 - A bag contains a white and b black balls. Balls...Ch. 3 - A round-robin tournament of n contestants is a...Ch. 3 - Prove directly thatP(EF)=P(EFG)P(GF)+P(EFGC)P(GCF)Ch. 3 - Prove the equivalence of Equations (5.11) and...Ch. 3 - Prob. 3.27TE Ch. 3 - Prove or give a counterexample, if E1 and E2 are...Ch. 3 - In Laplaces rule of succession (Example 5e ), show...Ch. 3 - In Laplaces rule of succession (Example 5e),...Ch. 3 - Prob. 3.31TE Ch. 3 - In a game of bridge, West has no aces What is the...Ch. 3 - Prob. 3.2STPE Ch. 3 - How can 20 balls, 10 white and 10 black, be put...Ch. 3 - Prob. 3.4STPE Ch. 3 - An urn has r red and w white balls that are...Ch. 3 - An urn contains b black balls and r red balls. One...Ch. 3 - A friend randomly chooses two cards, without...Ch. 3 - Show that P(HE)P(GE)=P(H)P(G)P(EH)P(EG). Suppose...Ch. 3 - You ask your neighbor to water a sickly plant...Ch. 3 - Six balls are to be randomly chosen from an urn...Ch. 3 - A type C battery is in working condition with...Ch. 3 - Prob. 3.12STPE Ch. 3 - Balls are randomly removed from an urn that...Ch. 3 - A coin having probability .8 of landing on heads...Ch. 3 - In a certain species of rats, black dominates over...Ch. 3 - a. In Problem 3.70b, find the probability that a...Ch. 3 - For the k-out-of-n system described in Problem...Ch. 3 - Prob. 3.18STPE Ch. 3 - Prob. 3.19STPE Ch. 3 - Suppose that there are n possible outcomes of a...Ch. 3 - If A flips vand B flips n fair coins, show that...Ch. 3 - Prove or give counterexamples to the following...Ch. 3 - Let A and B be events having positive probability....Ch. 3 - Rank the following from most likely to least...Ch. 3 - Two local factories, A and B, produce radios. Each...Ch. 3 - Show that if P(AB)=1, then P(BCAC)=1 Ch. 3 - Prob. 3.27STPE Ch. 3 - A total of 2n cards, of which 2 are aces, are to...Ch. 3 - There are n distinct types of coupons, and each...Ch. 3 - Show that for any events E and F,P(EEF)P(EF) Hint:...Ch. 3 - There is a 60 percent chance that event A will...

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