DISCRETE MATHEMATICS LOOSELEAF
8th Edition
ISBN: 9781264309689
Author: ROSEN
Publisher: MCG
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Chapter 7.2, Problem 10E
What is the probability of these events when we randomly select a permutation of the 26 lowercase letters of the English alphabet?
- The first 13 letters of the permutation are in alphabetical order.
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Consider the initial value problem
mx" + cx' + kx = F(t),
x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring system initially at rest and subjected to an
applied force F(t), where the unit of force is the Newton (N). Assume that m = = 2
kilograms, c = 8 kilograms per second, k 80 Newtons per meter, and F(t) = 20e¯*
=
Newtons.
Solve the initial value problem.
x(t) =
=
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0
t→∞
? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive
values of t.
For very large positive values of t,
x(t) ≈ x sp(t)
=
help (formulas)
Book: Section 2.6 of Notes on Diffy Qs
Consider the initial value problem
mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring system initially at rest and subjected to an
applied force F(t), where the unit of force is the Newton (N). Assume that m = 2
kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and
F(t) = 100 cos(8t) Newtons.
Solve the initial value problem.
x(t) =
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0
t→∞
? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive
values of t.
For very large positive values of t,
x(t)≈ x sp(t)
=
help (formulas)
Book: Section 2.6 of Notes on Diffy Qs
Chapter 7 Solutions
DISCRETE MATHEMATICS LOOSELEAF
Ch. 7.1 - i. What is the probability that a card selected at...Ch. 7.1 - t istheprobability that a fair die comes up six...Ch. 7.1 - t is the probability that a randomly selected...Ch. 7.1 - What is the probability7that a randomly selected...Ch. 7.1 - t is the probability that the sum of the numbers...Ch. 7.1 - t is the probability that a card selected at...Ch. 7.1 - t is the probability that when a coin is flipped...Ch. 7.1 - t is the probability that a five-card poker hand...Ch. 7.1 - t is the probability that a five-card poker hand...Ch. 7.1 - t is the probability that a five-card poker hand...
Ch. 7.1 - Prob. 11ECh. 7.1 - t is the probability that afive-card poker hand...Ch. 7.1 - t is the probability tliat afive-card poker hand...Ch. 7.1 - t istheprobability that a five-card poker hand...Ch. 7.1 - t is theprobabilifrthatafive-cardpoker hand...Ch. 7.1 - t is the probability7that a five-card poker hand...Ch. 7.1 - Prob. 17ECh. 7.1 - Mat is the probability' that a five-card poker...Ch. 7.1 - Prob. 19ECh. 7.1 - probabihh’thatafiM^Ch. 7.1 - Prob. 21ECh. 7.1 - t is the probability that a positive integer not...Ch. 7.1 - t is the probability that a positive integer not...Ch. 7.1 - Prob. 24ECh. 7.1 - - Find the probability of winning a lottery by...Ch. 7.1 - 26.Find the pr obabilitj- of selecting none of the...Ch. 7.1 - Prob. 27ECh. 7.1 - Prob. 28ECh. 7.1 - Prob. 29ECh. 7.1 - Prob. 30ECh. 7.1 - Prob. 31ECh. 7.1 - Prob. 32ECh. 7.1 - i$theprobabilitytiiatAbby,Barry,andSy^...Ch. 7.1 - 34.Mat is the probability' that Bo, Colleen, Jeff,...Ch. 7.1 - roulette, a wheel with 38 numbers is spun. Of...Ch. 7.1 - ch is more likely: rolling a total of 8 when two...Ch. 7.1 - ch is more likely: rolling a total of 9 when hvo...Ch. 7.1 - A player in the Mega Millions lottery picks five...Ch. 7.1 - a player buys a Mega Millions ticket in many...Ch. 7.1 - A player in the Powerball lottery picks five...Ch. 7.1 - Aplayer in the Powerball lottery (see Exercise 40)...Ch. 7.1 - Two events E i and E2are calledindependentifp(Etfl...Ch. 7.1 - Prob. 43ECh. 7.1 - Suppose that instead of three doors, there are...Ch. 7.1 - s problem was posed by the Chevalier de Mere and...Ch. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - w that conditions (2) and (22) are met under...Ch. 7.2 - A pair of dice is loaded. The probability that a 4...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - 8.What is the probability of these events when we...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - What is the probability of these events when we...Ch. 7.2 - pose, that £ and F are. events such that d(£)=0.7...Ch. 7.2 - pose that £ and Fare events such thatp(£) = 0.8...Ch. 7.2 - w that if £ and F are events, thenpfEn F) >p(E) +...Ch. 7.2 - Use mathematical induction to prove the following...Ch. 7.2 - w that if £x, £2,Enare events from afinite sample...Ch. 7.2 - Show that iff and f are independent events,...Ch. 7.2 - 17,It £ and F are independent events, prove or...Ch. 7.2 - What is the probability that hvo people chosen at...Ch. 7.2 - Mat is the probability that two people chosen at...Ch. 7.2 - Prob. 20ECh. 7.2 - Prob. 21ECh. 7.2 - February 29 occurs only inleap years, Years...Ch. 7.2 - ^Tiat is the conditional probabilitv that exactly...Ch. 7.2 - What is the. conditional probabilih' that exactly...Ch. 7.2 - Prob. 25ECh. 7.2 - Let Ebe the event that aranmly generated bit...Ch. 7.2 - Prob. 27ECh. 7.2 - a8. Assume that the probability a child is a boy...Ch. 7.2 - A group of six people play the game of “ odd...Ch. 7.2 - Find the probability that a randomly generated bit...Ch. 7.2 - Find the probability that a family with five...Ch. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Find each of the following probabilities...Ch. 7.2 - d each of the following probabilities...Ch. 7.2 - Prob. 36ECh. 7.2 - Prob. 37ECh. 7.2 - 38.A pair of dice is rolled in a remote location...Ch. 7.2 - This exercise employs the probabilistic method to...Ch. 7.2 - Dense a Monte Carlo algorithm that determines...Ch. 7.2 - pseudocode to write out the probabilistic...Ch. 7.3 - i.Suppose that £ andFare events in a sample space...Ch. 7.3 - Suppose that Land Fare events in a sample space...Ch. 7.3 - 3.Suppose that Frida selects a ball by first...Ch. 7.3 - 4.Suppo s e that Ann selects a ball by first...Ch. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - 8,Suppose that one person in 10,000 people has a...Ch. 7.3 - Suppose that 8% of the patients tested in a clinic...Ch. 7.3 - io,Suppose that 4% of the patients tested in a...Ch. 7.3 - ...Ch. 7.3 - ...Ch. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - In this exercise we will use Bayes' theorem to...Ch. 7.3 - Prob. 16ECh. 7.3 - Prob. 17ECh. 7.3 - 18.Suppose that a Bayesian spam filter is trained...Ch. 7.3 - 19,Suppose that a Bayesian spam filter is trained...Ch. 7.3 - Prob. 20ECh. 7.3 - ,Suppose that a Bayesian spam filter is trained on...Ch. 7.3 - Suppose that we have prior information concerning...Ch. 7.3 - Prob. 23ECh. 7.4 - t is the expected number of heads that come up...Ch. 7.4 - t is the expected number of heads that come up...Ch. 7.4 - t is the expected number of times a 6 appears when...Ch. 7.4 - A coin is biased so that the probability a head...Ch. 7.4 - ^Tiat is the expected sum of the numbers that...Ch. 7.4 - Prob. 6ECh. 7.4 - final exam of a discrete mathematics course...Ch. 7.4 - t is the expected sum of the numbers that appear...Ch. 7.4 - Prob. 9ECh. 7.4 - Suppose that we flip a fair coin until either it...Ch. 7.4 - Suppose that we roll a fair die until a 6 conies...Ch. 7.4 - pose that we roll a fair die until a 6 comes up....Ch. 7.4 - pose thatwerollapairoffair dice...Ch. 7.4 - Show that the sum of the probabilities of a random...Ch. 7.4 - Show that if the random variable A'has the...Ch. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Show that if J2,...,Xnare mutually independent...Ch. 7.4 - What is the expected value of the sum of the...Ch. 7.4 - as.Provethelaw of total expectations.Ch. 7.4 - Prob. 23ECh. 7.4 - Prob. 24ECh. 7.4 - A run is a maximal sequence of successes in a...Ch. 7.4 - a6.Let J(s) be a random variable, where I(s) is a...Ch. 7.4 - What is the variance of the number of heads that...Ch. 7.4 - t is the variance ot the number of times a 6...Ch. 7.4 - LetXnbe the random variable that equals the number...Ch. 7.4 - w that ifXand Fare independent random variables,...Ch. 7.4 - Prob. 31ECh. 7.4 - Pronde an example that shows that the variance of...Ch. 7.4 - pose that A\ andX2are independent Bernoulli trials...Ch. 7.4 - Prove the general caseofTheoremy. That is, show...Ch. 7.4 - Prob. 35ECh. 7.4 - Prob. 36ECh. 7.4 - Prob. 37ECh. 7.4 - pose that the number of cans of soda pop filled in...Ch. 7.4 - 39.Suppose that the number of aluminum cans...Ch. 7.4 - pose the probabilitvthatxis the...Ch. 7.4 - In this exercise we derive an estimate of the...Ch. 7.4 - Prob. 42ECh. 7.4 - to is the variance of the number of fixed...Ch. 7.4 - Prob. 44ECh. 7.4 - Prob. 45ECh. 7.4 - Prob. 46ECh. 7.4 - Prob. 47ECh. 7.4 - Prob. 48ECh. 7.4 - Prob. 49ECh. 7 - Define the probability of an event when all...Ch. 7 - WTiat conditions should be met by the...Ch. 7 - Define, the conditional probability’ of an event £...Ch. 7 - Prob. 4RQCh. 7 - tois a random variable? toare the possible values...Ch. 7 - Prob. 6RQCh. 7 - Explain how the average-case computational...Ch. 7 - Prob. 8RQCh. 7 - What does the linearity of expectations of random...Ch. 7 - Prob. 10RQCh. 7 - Prob. 11RQCh. 7 - Prob. 12RQCh. 7 - Prob. 13RQCh. 7 - What is the variance of the sum of n independent...Ch. 7 - Prob. 15RQCh. 7 - Prob. 1SECh. 7 - 2012, a player in the Mega Millions lottery picks...Ch. 7 - 2012, a player in the Powerball lottery picks five...Ch. 7 - t is the probability that a hand of 13 cards...Ch. 7 - t is the probability that a 13-card bridge hand...Ch. 7 - t is the probability that a seven-card poker hand...Ch. 7 - What is the expected value of the number that...Ch. 7 - What is the expected value of the number that...Ch. 7 - Suppose that a pair of fair octahedral dice is...Ch. 7 - io.Suppose that a pair offaiir dodecahedral diceis...Ch. 7 - ii.Supp o s e that a fair standard (cubic) die and...Ch. 7 - Prob. 12SECh. 7 - (mpeople!n>3!play“oddp™ut’todeadeMo^...Ch. 7 - Prob. 14SECh. 7 - posethatmandnarepositiYeintegers.Bat is...Ch. 7 - pose thatEt, E2,Enarenevents with p(£j) >o fori...Ch. 7 - Prob. 17SECh. 7 - t is the probability that when a fair coin is...Ch. 7 - t is the probability that a randomly selected bit...Ch. 7 - t is the probability that a randomly selected bit...Ch. 7 - sider the following game. A per son flips a coin...Ch. 7 - pose that n halls are tossed intobbins so that...Ch. 7 - posethatAandBareeventswthprobabilitiesp(A) =...Ch. 7 - posethat/l andB are events...Ch. 7 - all fromDefinition jinSection 7,2that the events...Ch. 7 - ...Ch. 7 - Prob. 27SECh. 7 - Prob. 28SECh. 7 - Prob. 29SECh. 7 - Prob. 30SECh. 7 - Prob. 31SECh. 7 - Prob. 32SECh. 7 - Prob. 33SECh. 7 - maximum satisfiability problemasks for an...Ch. 7 - Prob. 35SECh. 7 - The following method can be used to generate a...Ch. 7 - Prob. 1CPCh. 7 - Prob. 2CPCh. 7 - Prob. 3CPCh. 7 - Prob. 4CPCh. 7 - Prob. 5CPCh. 7 - ...Ch. 7 - Prob. 7CPCh. 7 - Prob. 8CPCh. 7 - Prob. 9CPCh. 7 - ulaterepeated trials oftheMoufr Hall Three-Door...Ch. 7 - Prob. 11CPCh. 7 - Prob. 1CAECh. 7 - Prob. 2CAECh. 7 - Prob. 3CAECh. 7 - Prob. 4CAECh. 7 - Prob. 5CAECh. 7 - Prob. 6CAECh. 7 - Prob. 7CAECh. 7 - Prob. 8CAECh. 7 - cribe the origins of probability theory and the...Ch. 7 - Prob. 2WPCh. 7 - 3.Discuss the probability' of winning when you...Ch. 7 - estigate the game of craps and discuss the...Ch. 7 - Prob. 5WPCh. 7 - Prob. 6WPCh. 7 - lain how Erdos and Renvi first used the...Ch. 7 - cuss the different types of probabilistic...
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- Consider the initial value problem mx" cx' + kx F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 80 Newtons per meter, and F(t) = 20 sin(6t) kilograms, c = 8 kilograms per second, k = Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 0047 ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ x sp(t) = ☐ help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = - 4xy with initial condition y(0) = 1.9. Recall that Runge-Kutta method has the following formula for computing the next step, where h is the step size: f(xi, Yi) = fx i + (++) k1 = h k2 2 ¯‚ Yi + k₁ h h k3 = fxi 2 `, Yi + k₂· 2 k4 = f(xi+h, yikзh) i+1=i+h k12k22k3 + k4 Yi+1 Yi + h 6 Using Runge-Kutta step size h = 0.4: Estimate y(0.4) ≈ help (numbers) Estimate y(0.8) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardDetermine which differential equation corresponds to each phase diagram. You should be able to state briefly how you know your choices are correct. х x 4 4 4 4 3 3 3 3 2 2 2 2 dx ? ✰ dt = 1. = x² - 3x 1 1 1 1 ? ◇ 2. dx dt = x(x − 2) - 0 0 0 0 ? ◇ 3. dx dt = x(2 − x)² -1 -1 -1 -1 Q -2 -2 -2 dx ? ◇ 4. ༤་ dt = = 3x - x² -3 -3 -3 -3 x³- 4x = x²|x − 2| ? ◇ 5. ம் dx dt བི་ dx ? ◇ 6. dt ཝེ་ dx ? 7. dt ཝེ་ dx ? ◇ 8. ཝེ་ dt -4 -4 -4 -4 A B 0 D = = 2x = x² * x * * x * K 4 4 4 4 = 4x - x³ 3 3 3 • 3 Book: Section 1.6 of Notes on Diffy Qs dit for this problem 2 2 2 2 1 1 1 1 0 0 0 8 -1 -1 -1 -1 N 心 -2 -2 -3 -3 -3 -4 -4 -4 -4 E FL G Harrow_forward
- Dear expert Chatgpt gives wrong answer Plz don't use chat gptarrow_forwardAn improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = xn +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution . İs Zn+1 = Yn + h · k₁. Then we find the slope at the predicted new point k₁ = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = = Yn + 1½ ½ (k1 + k₂). Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx = x - 0.5y, y(0.5) = 9. We let xo = 0.5 and yo 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k₂ 0 0.59-48 -3.25 ♡ <+ help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = = help (formulas) Thus the actual value of the…arrow_forwardAlready got wrong Chatgpt answer If ur also Chatgpt user leave itarrow_forward
- The graph of the function f(x) is 1,0 (the horizontal axis is x.) Consider the differential equation x' = f(x). List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equalibria are stable, semi-stable (stable from one side, unstable from the other), or unstable. x = help (numbers) x = help (numbers) x = help (numbers) x = help (numbers) Book: Section 1.6 of Notes on Diffy Qsarrow_forward= A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 60 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. Determine the spring constant k. k = Newtons/meter help (numbers) Hint is to use earth gravity of 9.8 meters per second squared, and note that Newton is kg meter per second squared. Formulate the initial value problem for x(t), where x(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. Give your answer in terms of x, x',x",t. Differential equation: | help (equations) Initial conditions: x (0) = and '(0) = help (numbers) Solve the initial value problem for x(t). x(t) = ☐ help (formulas) Plot the solution and determine the maximum displacement from equilibrium made by the object on the…arrow_forwardSuppose f(x) is a continuous function that is zero when x is −1, 3, or 6 and nowhere else. Suppose we tested the function at a few points and found that ƒ(−2) 0, and f(7) < 0. Let x(t) be the solution to x' f(x) and x(0) = 1. Compute: lim x(t) help (numbers) t→∞ Book: Section 1.6 of Notes on Diffy Qsarrow_forward
- Consider the initial value problem У y' = sin(x) + y(-4) = 5 4 Use Euler's Method with five steps to approximate y(-2) to at least two decimal places (but do not round intermediate results). y(-2) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = 5y with initial condition y(0) : The actual solution is y(1) = 207.78 help (numbers) = 1.4. We wish to analyze what happens to the error when estimating y(1) via Euler's method. Start with step size h = 1 (1 step). Compute y(1) Error 8.4 help (numbers) 199.38 help (numbers) Note: Remember that the error is the absolute value! Let us half the step size to h = 0.5 (2 steps). Compute y(1) ≈ 17.15 help (numbers) Error = 190.63 help (numbers) The error went down by the factor: Error Previous error Let us half the step size to h = 0.25 (4 steps). Compute y(1) 35.88046875 help (numbers) Error = 171.90 help (numbers) help (numbers) The error went down by the factor: Error Previous error help (numbers) Euler's method is a first order method so we expect the error to go down by a factor of 0.5 each halving. Of course, that's only very approximate, so the numbers you get above are not exactly 0.5. Book: Section 1.7 of Notes on Diffy Qsarrow_forwardAnswer all the boxes and box the answers. Thank you write it downarrow_forward
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