 8.3.1: If A and B are two disjoint sets, then __________.
 8.3.2: If and are two nonempty, mutually exclusive events in a sample spac...
 8.3.3: True or False If and form a partition of a sample space S and if E ...
 8.3.4: True or False If and form a partition of a sample space S and if E ...
 8.3.5: In 518, find the indicated probabilities by referring to the tree d...
 8.3.6: In 518, find the indicated probabilities by referring to the tree d...
 8.3.7: In 518, find the indicated probabilities by referring to the tree d...
 8.3.8: In 518, find the indicated probabilities by referring to the tree d...
 8.3.9: In 518, find the indicated probabilities by referring to the tree d...
 8.3.10: In 518, find the indicated probabilities by referring to the tree d...
 8.3.11: In 518, find the indicated probabilities by referring to the tree d...
 8.3.12: In 518, find the indicated probabilities by referring to the tree d...
 8.3.13: In 518, find the indicated probabilities by referring to the tree d...
 8.3.14: In 518, find the indicated probabilities by referring to the tree d...
 8.3.15: In 518, find the indicated probabilities by referring to the tree d...
 8.3.16: In 518, find the indicated probabilities by referring to the tree d...
 8.3.17: In 518, find the indicated probabilities by referring to the tree d...
 8.3.18: In 518, find the indicated probabilities by referring to the tree d...
 8.3.19: Events and form a partition of a sample space S with and If E is an...
 8.3.20: Events and form a partition of a sample space S with and If E is an...
 8.3.21: Events and form a partition of a sample space S with and If E is an...
 8.3.22: Events and form a partition of a sample space S with and If P(A1) =...
 8.3.23: Use the information in to find and
 8.3.24: Use the information in to find and
 8.3.25: Use the information in to find and
 8.3.26: Use the information in to find and P(A3 E).
 8.3.27: Three jars contain colored balls as follows: Jar Red, R White, W Bl...
 8.3.28: Car Production Cars are being produced by two factories, but factor...
 8.3.29: Color Blindness According to the 2000 U.S. Census, 50.9% of the U.S...
 8.3.30: Hospital Bills A hospital billing department knows that the probabi...
 8.3.31: Advertising Castaway Cruise lines mails a promotional advertisement...
 8.3.32: Market Research From past experience Kave Jewelers know that 80% of...
 8.3.33: OnTime Performance You are meeting a friend at Fort Myers Airport....
 8.3.34: Air Travel Liz flies regularly from Midway Airport in Chicago to Ha...
 8.3.35: Customer Surveys Juan Morales is a restauranteur who owns three aut...
 8.3.36: Female Unemployment Based on information from the Bureau of Labor S...
 8.3.37: Individual Tax Audits In 2008, the audit risk for the average indiv...
 8.3.38: Promotional Coupon An office supply store desires to increase sales...
 8.3.39: Occupations Based on data from the Bureau of Labor Statistics, 36.3...
 8.3.40: Medical Diagnosis In a certain small town, 16% of the population de...
 8.3.41: Voting Pattern In Cook County, 55% of the registered voters are Dem...
 8.3.42: Quality Control A computer manufacturer has three assembly plants. ...
 8.3.43: Oil Drilling An oil well is to be drilled in a certain location. Th...
 8.3.44: Oil Drilling A geologist is using seismographs to test for oil. It ...
 8.3.45: Political Polls In conducting a political poll, a pollster divides ...
 8.3.46: TB Screening Suppose that if a person with tuberculosis is given a ...
 8.3.47: Detective Columbo An absentminded nurse is to give Mr. Brown a pil...
 8.3.48: Marketing To introduce a new beer, a company conducted a survey. It...
 8.3.49: Car Insurance and Age Insurers know that young drivers between the ...
 8.3.50: Medical Test A scientist designed a medical test for a certain dise...
 8.3.51: Testing for HIV An article in the New York Times some time ago repo...
 8.3.52: Test for Tuberculosis A faster and simpler diagnostic test for the ...
 8.3.53: Show that if F is a subset of E and
 8.3.54: Prove Bayes Theorem, Formula (8).
Solutions for Chapter 8.3: Bayes Theorem
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach  11th Edition
ISBN: 9780470876398
Solutions for Chapter 8.3: Bayes Theorem
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.