A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 3, Problem 3.83P
In a certain contest, the players are of equal skill and the
A plays in exactly i contests
E: A and B never play each other
- Find P(Ai),i=1,...,n.
- Find P(E).
- Let Pn=P(E).
- Show that Pn=12n−1+2n+22n−1(12)2Pn−1 are use this formula to check the answer you obtained in part (b).
Hint: Find P(E) by conditioning on which of the events P(Ai),i=1,...,n occur. In simplifying your answer, use the algebraic identity n−1∑i=1ixi−1=1−nxn−1+(n−1)xn(1−x)2
For another approach to solving this problem, note that there are a total of 2n−1 games played.
- Explain why 2n−1 games are played.
Number these games, and let
Bi denote
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Chapter 3 Solutions
A First Course in Probability (10th Edition)
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 - Suppose distinct values are written on each of 3...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 - Twenty percent of Bs phone calls are with her...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 - There is a 30 percent chance that A can fix her...Ch. 3 - In any given year, a male automobile policyholder...Ch. 3 - An urn contains 5 white and 10 black balls. A fair...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 - Players A, B, C, D are randomly lined up. The...Ch. 3 - Players 1,2,3 are playing a tournament. Two of...Ch. 3 - Suppose there are two coins, with coin 1 landing...Ch. 3 - In a 7 game series played with two teams, the...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 - 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 50—50 chance that the queen carries...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. 3.79PCh. 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 an eight team tournament with the format...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 - Each of n workers is independently qualified to do...Ch. 3 - Suppose in the preceding problem that n=2 and that...Ch. 3 - Each member of a population of size n is,...Ch. 3 - Show that if P(A)0, then P(ABA)P(ABAB)Ch. 3 - Prob. 3.2TECh. 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 - Independent trials that result in a success with...Ch. 3 - Suppose that n independent trials are performed,...Ch. 3 - Let Q. denote the probability that no run of 3...Ch. 3 - Consider the gamblers ruin problem, with the...Ch. 3 - Prob. 3.20TECh. 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. 3.27TECh. 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 - Suppose that a nonmathematical, but...Ch. 3 - In a game of bridge, West has no aces What is the...Ch. 3 - Prob. 3.2STPECh. 3 - How can 20 balls, 10 white and 10 black, be put...Ch. 3 - Prob. 3.4STPECh. 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. 3.12STPECh. 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. 3.18STPECh. 3 - Prob. 3.19STPECh. 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)=1Ch. 3 - Prob. 3.27STPECh. 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 - a. If the odds of A is 23, what is the probability...Ch. 3 - Prob. 3.32STPECh. 3 - If the events E, F, G are independent. show that...Ch. 3 - Players 1,2,3, are in a contest. Two of them are...Ch. 3 - If 4 balls are randomly chosen from an urn...Ch. 3 - In a 4 player tournament, player 1 plays player 2,...Ch. 3 - In a tournament Involving players 1,..., n,...
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