 12.4.12.1.181: The expected gain or loss of an experiment over the long run is cal...
 12.4.12.1.182: In an experiment, if an individual expects to have a loss in the lo...
 12.4.12.1.183: In an experiment, if an individual expects to have a gain in the lo...
 12.4.12.1.184: In an experiment, if an individual expects to break even in the lon...
 12.4.12.1.185: Expected Attendance For an outdoor art and craft show, event organi...
 12.4.12.1.186: New Business In a proposed business venture, Stephanie Morrison est...
 12.4.12.1.187: Basketball Shenise Johnson is a star player for the University of M...
 12.4.12.1.188: Career Fair Attendance For a Nursing and Allied Health Care Career ...
 12.4.12.1.189: TV Shows The Fox television network is scheduling its fall lineup o...
 12.4.12.1.190: Seattle Greenery In July in Seattle, the grass grows 1 2 in. a day ...
 12.4.12.1.191: Investment Club The Triple L investment club is considering purchas...
 12.4.12.1.192: Clothing Sale At a special clothing sale at the Crescent Oaks Count...
 12.4.12.1.193: Fortune Cookies At the Royal Dragon Chinese restaurant, a slip in t...
 12.4.12.1.194: Pick a Card Mike and Dave play the following game. Mike picks a car...
 12.4.12.1.195: Roll a Die Alyssa and Gabriel play the following game. Alyssa rolls...
 12.4.12.1.196: Blue Chips and Red Chips A bag contains 4 blue chips and 6 red chip...
 12.4.12.1.197: MultipleChoice Test A multiplechoice exam has five possible answe...
 12.4.12.1.198: MultipleChoice Test A multiplechoice exam has four possible answe...
 12.4.12.1.199: Fair Price The expected value when you purchase a lottery ticket is...
 12.4.12.1.200: Fair Price The expected value of a carnival game is $6.50, and the...
 12.4.12.1.201: Raffle Tickets Five hundred raffle tickets are sold for $3 each. On...
 12.4.12.1.202: Raffle Tickets One thousand raffle tickets are sold for $1 each. On...
 12.4.12.1.203: Raffle Tickets Two thousand raffle tickets are sold for $3 each. Th...
 12.4.12.1.204: Raffle Tickets Ten thousand raffle tickets are sold for $5 each. Fo...
 12.4.12.1.205: Spinners In Exercises 25 and 26, assume that a person spins the poi...
 12.4.12.1.206: Spinners In Exercises 25 and 26, assume that a person spins the poi...
 12.4.12.1.207: Spinners In Exercises 27 and 28, assume that a person spins the poi...
 12.4.12.1.208: Spinners In Exercises 27 and 28, assume that a person spins the poi...
 12.4.12.1.209: Selecting an Envelope In Exercises 2932, a person randomly selects ...
 12.4.12.1.210: Selecting an Envelope In Exercises 2932, a person randomly selects ...
 12.4.12.1.211: Selecting an Envelope In Exercises 2932, a person randomly selects ...
 12.4.12.1.212: Selecting an Envelope In Exercises 2932, a person randomly selects ...
 12.4.12.1.213: Spinners In Exercises 3336, assume that a person spins the pointer ...
 12.4.12.1.214: Spinners In Exercises 3336, assume that a person spins the pointer ...
 12.4.12.1.215: Spinners In Exercises 3336, assume that a person spins the pointer ...
 12.4.12.1.216: Spinners In Exercises 3336, assume that a person spins the pointer ...
 12.4.12.1.217: Selecting an Envelope In Exercises 37 40 on page 706, a person rand...
 12.4.12.1.218: Selecting an Envelope In Exercises 37 40 on page 706, a person rand...
 12.4.12.1.219: Selecting an Envelope In Exercises 37 40 on page 706, a person rand...
 12.4.12.1.220: Selecting an Envelope In Exercises 37 40 on page 706, a person rand...
 12.4.12.1.221: Reaching Base Safely Based on his past baseball history, Jim Devias...
 12.4.12.1.222: Life Insurance According to Bristol Mutual Life Insurances mortalit...
 12.4.12.1.223: Choosing a Colored Chip In a box, there are a total of 10 chips. Th...
 12.4.12.1.224: Choosing a Colored Chip Repeat Exercise 43 but assume that an orang...
 12.4.12.1.225: Employee Hiring The academic vice president at Brookdale Community ...
 12.4.12.1.226: Completing a Project A mechanical contractor is preparing for a con...
 12.4.12.1.227: New Store Dunkin Donuts is opening a new store. The company estimat...
 12.4.12.1.228: China Cabinet The owner of an antique store estimates that there is...
 12.4.12.1.229: Rolling a Die A die is rolled many times, and the points facing up ...
 12.4.12.1.230: Lawsuit Don Vello is considering bringing a lawsuit against the Dum...
 12.4.12.1.231: Road Service On a clear day in Boston, the Automobile Association o...
 12.4.12.1.232: Real Estate The expenses for Jorge Estrada, a real estate agent, to...
 12.4.12.1.233: Dart Board In Exercises 53 and 54, assume that you are blindfolded ...
 12.4.12.1.234: Dart Board In Exercises 53 and 54, assume that you are blindfolded ...
 12.4.12.1.235: Term Life Insurance An insurance company will pay the face value of...
 12.4.12.1.236: Lottery Ticket Is it possible to determine your expectation when yo...
 12.4.12.1.237: Roulette In Exercises 57 and 58, use the roulette wheel illustrated...
 12.4.12.1.238: Roulette In Exercises 57 and 58, use the roulette wheel illustrated...
 12.4.12.1.239: Wheel of Fortune The following is a miniature version of the Wheel ...
Solutions for Chapter 12.4: Probability
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 12.4: Probability
Get Full SolutionsChapter 12.4: Probability includes 59 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 59 problems in chapter 12.4: Probability have been answered, more than 70856 students have viewed full stepbystep solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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].

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.