 12.887: In your own words, explain the law of large numbers.
 12.888: Explain how empirical probability can be used to determine whether ...
 12.889: Cars Of 45 people who purchase a vehicle at a car dealership, 9 pur...
 12.890: Cards Select a card from a deck of cards 40 times with replacement ...
 12.891: Television News In a small town, 250 people were asked whether they...
 12.892: Digits In Exercises 69, each of the digits 0, 1, 2, 3, 4, 5, 6, 7, ...
 12.893: Digits In Exercises 69, each of the digits 0, 1, 2, 3, 4, 5, 6, 7, ...
 12.894: Digits In Exercises 69, each of the digits 0, 1, 2, 3, 4, 5, 6, 7, ...
 12.895: Digits In Exercises 69, each of the digits 0, 1, 2, 3, 4, 5, 6, 7, ...
 12.896: Yogurt Preference In Exercises 1013, a taste test is given to 50 cu...
 12.897: Yogurt Preference In Exercises 1013, a taste test is given to 50 cu...
 12.898: Yogurt Preference In Exercises 1013, a taste test is given to 50 cu...
 12.899: Yogurt Preference In Exercises 1013, a taste test is given to 50 cu...
 12.900: Obesity According to the U.S. Centers for Disease Control and Preve...
 12.901: Vegetable Mixup Nicholas Delaney, a mischievous little boy, has re...
 12.902: Horseracing The odds against Buttermilk winning the Triple Crown in...
 12.903: Dry Cleaner Success The probability that a new dry cleaner will suc...
 12.904: Raffle Tickets One thousand raffle tickets are sold at $2 each. Thr...
 12.905: Expectation of a Card If Cameron selects a picture card from a stan...
 12.906: Expected Attendance If the day is sunny, 1000 people will attend th...
 12.907: Fair Price At a game of chance, the expected value is found to be ...
 12.908: Fair Price The expected value when you purchase a lottery ticket is...
 12.909: Club Officers Tina, Jake, Gina, and Carla form a club. They plan to...
 12.910: A Coin and a Number A coin is flipped and then a number from 1 thro...
 12.911: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.912: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.913: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.914: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.915: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.916: Spinning Two Wheels In Exercises 2530, the outer and inner wheels a...
 12.917: Sports Drink Selection In Exercises 3134, assume that Skip Bailey i...
 12.918: Sports Drink Selection In Exercises 3134, assume that Skip Bailey i...
 12.919: Sports Drink Selection In Exercises 3134, assume that Skip Bailey i...
 12.920: Sports Drink Selection In Exercises 3134, assume that Skip Bailey i...
 12.921: Spinner Probabilities In Exercises 3538, assume that the spinner ca...
 12.922: Spinner Probabilities In Exercises 3538, assume that the spinner ca...
 12.923: Spinner Probabilities In Exercises 3538, assume that the spinner ca...
 12.924: Spinner Probabilities In Exercises 3538, assume that the spinner ca...
 12.925: Spinner Probabilities In Exercises 3942, assume that the spinner ca...
 12.926: Spinner Probabilities In Exercises 3942, assume that the spinner ca...
 12.927: Spinner Probabilities In Exercises 3942, assume that the spinner ca...
 12.928: Spinner Probabilities In Exercises 3942, assume that the spinner ca...
 12.929: Restaurant Service In Exercises 43 46, use the results of a survey ...
 12.930: Restaurant Service In Exercises 43 46, use the results of a survey ...
 12.931: Restaurant Service In Exercises 43 46, use the results of a survey ...
 12.932: Restaurant Service In Exercises 43 46, use the results of a survey ...
 12.933: Neuroscience In Exercises 4750, assume that in a neuroscience cours...
 12.934: Neuroscience In Exercises 4750, assume that in a neuroscience cours...
 12.935: Neuroscience In Exercises 4750, assume that in a neuroscience cours...
 12.936: Neuroscience In Exercises 4750, assume that in a neuroscience cours...
 12.937: Television Show Four contestants are on a television show. There ar...
 12.938: Spelling Bee Five finalists remain in a high school spelling bee. T...
 12.939: Candy Selection Mrs. Williams takes her 3 children shopping. Each o...
 12.940: Astronaut Selection Three of nine astronauts must be selected for a...
 12.941: Vaccine Dr. Goldberg has three doses of vaccine for influenza type ...
 12.942: Dogsled Ten huskies are to be selected to pull a dogsled. How many ...
 12.943: Mega Millions To play the lottery game Mega Millions, you select 5 ...
 12.944: ParentTeacher Committee A committee of 7 is to be formed from 10 p...
 12.945: Selecting Test Subjects In a psychology research laboratory, one ro...
 12.946: Choosing Two Aces Two cards are selected at random, without replace...
 12.947: Color Chips In Exercises 61 64, a bag contains five red chips, thre...
 12.948: Color Chips In Exercises 61 64, a bag contains five red chips, thre...
 12.949: Color Chips In Exercises 61 64, a bag contains five red chips, thre...
 12.950: Color Chips In Exercises 61 64, a bag contains five red chips, thre...
 12.951: Magazines In Exercises 65 68, on a table in a doctors office are fi...
 12.952: Magazines In Exercises 65 68, on a table in a doctors office are fi...
 12.953: Magazines In Exercises 65 68, on a table in a doctors office are fi...
 12.954: Magazines In Exercises 65 68, on a table in a doctors office are fi...
 12.955: New Homes In the community of Spring Hill, 60% of the homes purchas...
 12.956: LongStemmed Roses At the Floyds Flower Shop, 1 5 of people orderin...
 12.957: Taking a Math Course During any semester at City College, 60% of th...
 12.958: Flowers Of the last 50 people who purchased flowers at Whistlestop ...
 12.959: One Sheet of Paper In Exercises 2 4, each of the numbers 19 is writ...
 12.960: One Sheet of Paper In Exercises 2 4, each of the numbers 19 is writ...
 12.961: One Sheet of Paper In Exercises 2 4, each of the numbers 19 is writ...
 12.962: Two Sheets of Paper In Exercises 57, if two of the same nine sheets...
 12.963: Two Sheets of Paper In Exercises 57, if two of the same nine sheets...
 12.964: Two Sheets of Paper In Exercises 57, if two of the same nine sheets...
 12.965: Two Sheets of Paper In Exercises 57, if two of the same nine sheets...
 12.966: One Chip and One Die In Exercises 913, one colored chipred, blue, o...
 12.967: One Chip and One Die In Exercises 913, one colored chipred, blue, o...
 12.968: In Exercises 1113, by observing the sample space of the chips and d...
 12.969: In Exercises 1113, by observing the sample space of the chips and d...
 12.970: In Exercises 1113, by observing the sample space of the chips and d...
 12.971: Passwords A personal password for an Internet brokerage account is ...
 12.972: Puppies A litter of collie puppies consists of four males and five ...
 12.973: Pick a Card You get to select one card at random from a standard de...
 12.974: Cars and SUVs The number of cars and the number of SUVs going throu...
 12.975: Awarding Prizes Three of six people are to be selected and given sm...
 12.976: Quality Control A bin contains a total of 20 batteries, of which 6 ...
 12.977: University Admission The probability that a person is accepted for ...
 12.978: Your cars speedometer indicates that you are traveling at 65 mph. W...
 12.979: A 10question multiplechoice exam is given, and each question has ...
 12.980: Many cars have keyless entry. To open the lock, you may press a 5d...
Solutions for Chapter 12: Probability
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 12: Probability
Get Full SolutionsChapter 12: Probability includes 94 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Since 94 problems in chapter 12: Probability have been answered, more than 78948 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.