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 12.1.12.1.1: A controlled operation that yields a set of results is called a(n) .
 12.1.12.1.2: The possible results of an experiment are called its .
 12.1.12.1.3: A subcollection of the outcomes of an experiment is called a(n) .
 12.1.12.1.4: Probability is classified as either theoretical probability or prob...
 12.1.12.1.5: Probability determined by the relative frequency of occurrence of a...
 12.1.12.1.6: Probability determined through a study of the possible outcomes tha...
 12.1.12.1.7: Flip a Coin Flip a coin 50 times and record the results. Determine ...
 12.1.12.1.8: Pair of Dice Roll a pair of dice 60 times and record the sums. Dete...
 12.1.12.1.9: Roll a Die Roll a die 50 times and record the results. Determine th...
 12.1.12.1.10: Two Coins Flip two coins 50 times and record the number of times ex...
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 12.1.12.1.16: Travel Web Sites The following table shows the number of visitors t...
 12.1.12.1.17: TopSelling Video Games The following table shows the number of vid...
 12.1.12.1.18: Grade Distribution Mr. Dooles grade distribution over the past 3 ye...
 12.1.12.1.19: Election In an election for student council president at Russell Sa...
 12.1.12.1.20: Volunteer Hours The graph below shows the average number of hours v...
 12.1.12.1.21: Hitting a BullsEye The pattern of hits shown on the target resulte...
 12.1.12.1.22: Rock Toss Jim Handy finds an irregularly shaped fivesided rock. He...
 12.1.12.1.23: Cell Biology Experiment An experimental serum was injected into 500...
 12.1.12.1.24: Gender In a large lecture hall there are 325 students of which 160 ...
 12.1.12.1.25: Mendels Experiment In one of his experiments (see pages 677678), Me...
 12.1.12.1.26: SecondGeneration Offspring In another experiment, Mendel crossbred...
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Solutions for Chapter 12.1: Probability
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 12.1: Probability
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 82 problems in chapter 12.1: Probability have been answered, more than 71116 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. Chapter 12.1: Probability includes 82 full stepbystep solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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(D» O.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.