 7.2.27E: Let E and F be the events that a family of n children has children ...
 7.2.25E: What is the conditional probability that a randomly generated bit s...
 7.2.29E: A group of six people play the game of “odd person out” to determin...
 7.2.30E: Find the probability that a randomly generated bit string of length...
 7.2.31E: Find the probability that a family with five children does not have...
 7.2.33E: Find the probability that the first child of a family with five chi...
 7.2.32E: Find the probability that a randomly generated bit string of length...
 7.2.34E: Find each of the following probabilities when n independent Bernoul...
 7.2.35E: Find each of the following probabilities when n independent Bernoul...
 7.2.36E: Use mathematical induction to prove that if E1, E2,…,En is a sequen...
 7.2.39E: This exercise employs the probabilistic method to prove a result ab...
 7.2.38E: A pair of dice is rolled in a remote location and when you ask an h...
 7.2.37E: (Requires calculus) Show that if E1, E2,…,En is an infinite sequenc...
 7.2.13E: Show that if E and F are events, then p(E ? F) ? p(E) + p(F) –1. Th...
 7.2.40E: Devise a Monte Carlo algorithm that determines whether a permutatio...
 7.2.41E: Use pseudocode to write out the probabilistic primality test descri...
 7.2.1E: What probability should be assigned to the outcome of heads when a ...
 7.2.3E: Find the probability of each outcome when a biased die is rolled. i...
 7.2.2E: Find the probability of each outcome when a loaded die is rolled. i...
 7.2.4E: Show that conditions (i) and (ii) are met under Laplace’s definitio...
 7.2.5E: A pair of dice is loaded. The probability that a 4 appears on the f...
 7.2.6E: What is the probability of these events when we randomly select a p...
 7.2.7E: What is the probability of these events when we randomly select a p...
 7.2.9E: What is the probability of these events when we randomly select a p...
 7.2.10E: What is the probability of these events when we randomly select a p...
 7.2.11E: Suppose that E and F are events such that p(E) = 0.7 and p(F) = 0.5...
 7.2.12E: Suppose that E and F are events such that p(E) = 0.8 and p(F) = 0.6...
 7.2.8E: What is the probability of these events when we randomly select a p...
 7.2.15E: Show that if E1, E2,…, En are events from a finite sample space, th...
 7.2.14E: Use mathematical induction to prove the following generalization of...
 7.2.18E: a) What is the probability that two people chosen at random were bo...
 7.2.16E: Show that if E and F are independent events, then and are also inde...
 7.2.17E: If E and F are independent events, prove or disprove that and are n...
 7.2.19E: a) What is the probability that two people chosen at ran dom were b...
 7.2.21E: Find the smallest number of people you need to choose at random so ...
 7.2.20E: Find the smallest number of people you need to choose at random so ...
 7.2.23E: What is the conditional probability that exactly fourheads appear w...
 7.2.22E: February 29 occurs only in leap years. Years divisible by 4, but no...
 7.2.24E: What is the conditional probability that exactly four heads appear ...
 7.2.28E: Assume that the probability a child is a boy is 0.51 and that the s...
 7.2.26E: Let E be the event that a randomly generated bit string of length t...
Solutions for Chapter 7.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 7.2
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 41 problems in chapter 7.2 have been answered, more than 198578 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 7.2 includes 41 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.