 8.2.1: True or False If E and F are mutually exclusive, then they are also...
 8.2.2: If E and F are two independent events, then P(E F) = .
 8.2.3: If E and F are independent events and if and find
 8.2.4: If E and F are independent events and if and find P(F).
 8.2.5: If E and F are independent events, find P(F) if and
 8.2.6: If E and F are independent events, find P(E) if and
 8.2.7: Suppose E and F are two events such that and Are E and F independent?
 8.2.8: If E and F are two events such that and are E and F independent?
 8.2.9: If E and F are two independent events with and
 8.2.10: If E and F are independent events with and find (a) (b) (c) (d)
 8.2.11: If E, F, and G are three independent events with and find
 8.2.12: If and are four independent events with and find
 8.2.13: If and what is Are E and F independent?
 8.2.14: If and what is P(E F)? Are E and F independent?
 8.2.15: Tmaze In a Tmaze a mouse may turn to the right (R) and receive a ...
 8.2.16: Drawing Cards A first card is drawn at random from a regular deck o...
 8.2.17: Selecting Marbles A box has 10 marbles in it, 6 red and 4 white. Su...
 8.2.18: Survey In a survey of 100 people, categorized as drinkers or nondri...
 8.2.19: Cardiovascular Disease Records show that a child of parents with he...
 8.2.20: Sex of Newborns The probability of a newborn baby being a girl is 0...
 8.2.21: Germination In a group of seeds, of which should produce violets, t...
 8.2.22: Insurance By examining the past driving records of 840 randomly sel...
 8.2.23: Quality Control Efraim Furniture Manufacturing Company hires two pe...
 8.2.24: AARP Life Insurance Underwriters for AARP sell life insurance to an...
 8.2.25: Stock Selection As a stockbroker, you recommend two stocks to a cli...
 8.2.26: Earnings and a Bachelors Degree According to the U.S. Census Bureau...
 8.2.27: Health Insurance The contingency table representing the numbers, in...
 8.2.28: Women in Business The data below represent the number of womenowne...
 8.2.29: Pumping Station A pumping station at a hydroelectric plant operates...
 8.2.30: Ambulatory Care The following table represents the number of ambula...
 8.2.31: Property Crime In a survey of 500 property crimes, the following da...
 8.2.32: Voting Patterns The following data show the number of voters in a s...
 8.2.33: Election A candidate for office believes that of registered voters ...
 8.2.34: Keys Selection A woman has 10 keys but only 1 fits her door. She tr...
 8.2.35: Testing Components (a) Create a table like Table 3 if the probabili...
 8.2.36: Testing Components Compute the expected number of tests saved per c...
 8.2.37: Allergy Testing A persons blood needs to be tested for an allergic ...
 8.2.38: Chevalier de Meres of the following random events do you think is m...
 8.2.39: Show that whenever two events are both independent and mutually exc...
 8.2.40: Let E be any event. If F is an impossible event, show that E and F ...
 8.2.41: Show that if E and F are independent events, so are and [Hint: Use ...
 8.2.42: Show that if E and F are independent events and if then E and F are...
 8.2.43: Suppose and E is independent of F. Show that F is independent of E....
 8.2.44: Prove Formula (1) on page 435. [Hint: If E and F are independent ev...
 8.2.45: Give examples of two events that are (a) Independent but not mutual...
Solutions for Chapter 8.2: Independent Events
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach  11th Edition
ISBN: 9780470876398
Solutions for Chapter 8.2: Independent Events
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Finite Mathematics, Binder Ready Version: An Applied Approach was written by and is associated to the ISBN: 9780470876398. This textbook survival guide was created for the textbook: Finite Mathematics, Binder Ready Version: An Applied Approach, edition: 11. Since 45 problems in chapter 8.2: Independent Events have been answered, more than 16155 students have viewed full stepbystep solutions from this chapter. Chapter 8.2: Independent Events includes 45 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.