A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 4, Problem 4.47P
In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with
a. What is the probability that a guilty defendant is declared guilty when there are (i) 9, (ii) 8, and (iii) 7 judges?
b. Repeat part (a) for an innocent defendant.
c. If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60 percent certain that the client is guilty?
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Q1. A group of five applicants for a pair of identical jobs consists of three men and two
women. The employer is to select two of the five applicants for the jobs. Let S
denote the set of all possible outcomes for the employer's selection. Let A denote
the subset of outcomes corresponding to the selection of two men and B the subset
corresponding to the selection of at least one woman. List the outcomes in A, B,
AUB, AN B, and An B. (Denote the different men and women by M₁, M2, M3
and W₁, W2, respectively.)
Q3 (8 points)
Q3. A survey classified a large number of adults according to whether they were diag-
nosed as needing eyeglasses to correct their reading vision and whether they use
eyeglasses when reading. The proportions falling into the four resulting categories
are given in the following table:
Use Eyeglasses for Reading
Needs glasses Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from the large group, find the probabilities of the events
defined below. The adult
(a) needs glasses.
(b) needs glasses but does not use them.
(c) uses glasses whether the glasses are needed or not.
4. (i) Let a discrete sample space be given by
N = {W1, W2, W3, W4},
and let a probability measure P on be given by
P(w1) = 0.2, P(w2) = 0.2, P(w3) = 0.5, P(wa) = 0.1.
Consider the random variables X1, X2 → R defined by
X₁(w1) = 1, X₁(w2) = 2,
X2(w1) = 2, X2 (w2) = 2,
Find the joint distribution of X1, X2.
(ii)
X1(W3) = 1, X₁(w4) = 1,
X2(W3) = 1, X2(w4) = 2.
[4 Marks]
Let Y, Z be random variables on a probability space (, F, P).
Let the random vector (Y, Z) take on values in the set [0, 1] x [0,2] and let the
joint distribution of Y, Z on [0, 1] x [0,2] be given by
1
dPy,z (y, z) ==(y²z+yz2) dy dz.
harks 12 Find the distribution Py of the random variable Y.
[8 Marks]
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...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.Similar questions
- marks 11 3 3/4 x 1/4 1. There are 4 balls in an urn, of which 3 balls are white and 1 ball is black. You do the following: draw a ball from the urn at random, note its colour, do not return the ball to the urn; draw a second ball, note its colour, return the ball to the urn; finally draw a third ball and note its colour. (i) Describe the corresponding discrete probability space (Q, F, P). [9 Marks] (ii) Consider the following event, A: Among the first and the third balls, one ball is white, the other is black. Write down A as a subset of the sample space and find its probability, P(A). [2 Marks]arrow_forwardThere are 4 balls in an urn, of which 3 balls are white and 1 ball isblack. You do the following:• draw a ball from the urn at random, note its colour, do not return theball to the urn;• draw a second ball, note its colour, return the ball to the urn;• finally draw a third ball and note its colour.(i) Describe the corresponding discrete probability space(Ω, F, P). [9 Marks](ii) Consider the following event,A: Among the first and the third balls, one ball is white, the other is black.Write down A as a subset of the sample space Ω and find its probability, P(A)arrow_forwardLet (Ω, F, P) be a probability space and let X : Ω → R be a randomvariable whose probability density function is given by f(x) = 12 |x|e−|x| forx ∈ R.(i) Find the characteristic function of the random variable X.[8 Marks](ii) Using the result of (i), calculate the first two moments of therandom variable X, i.e., E(Xn) for n = 1, 2. [6 Marks]Total marks 16 (iii) What is the variance of X?arrow_forward
- ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rulearrow_forwardLet X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .arrow_forwardLet P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find itarrow_forward
- 13) Consider the checkerboard arrangement shown below. Assume that the red checker can move diagonally upward, one square at a time, on the white squares. It may not enter a square if occupied by another checker, but may jump over it. How many routes are there for the red checker to the top of the board?arrow_forward12) The prime factors of 1365 are 3, 5, 7 and 13. Determine the total number of divisors of 1365.arrow_forward11) What is the sum of numbers in row #8 of Pascal's Triangle?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Bayes' Theorem 1: Introduction and conditional probability; Author: Dr Nic's Maths and Stats;https://www.youtube.com/watch?v=lQVkXfJ-rpU;License: Standard YouTube License, CC-BY
What is Conditional Probability | Bayes Theorem | Conditional Probability Examples & Problems; Author: ACADGILD;https://www.youtube.com/watch?v=MxOny_1y2Q4;License: Standard YouTube License, CC-BY
Bayes' Theorem of Probability With Tree Diagrams & Venn Diagrams; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=OByl4RJxnKA;License: Standard YouTube License, CC-BY
Bayes' Theorem - The Simplest Case; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=XQoLVl31ZfQ;License: Standard Youtube License