 6.2.6.2.1: What probability should be assigned to the outcome of heads when a ...
 6.2.6.2.2: Find the probability of each outcome when a loaded die is rolled, i...
 6.2.6.2.3: Find the probability of each outcome when a biased die is rolled, i...
 6.2.6.2.4: Show that conditions (i) and (ii) are met under Laplace's definitio...
 6.2.6.2.5: A pair of dice is loaded. The probability that a 4 appears on the f...
 6.2.6.2.6: What is the probability ofthese events when we randomly select a pe...
 6.2.6.2.7: What is the probability ofthese events when we randomly select a pe...
 6.2.6.2.8: What is the probability of these events when we randomly select a p...
 6.2.6.2.9: What is the probability of these events when we randomly select a p...
 6.2.6.2.10: What is the probability ofthese events when we randomly select a pe...
 6.2.6.2.11: Suppose that E and F are events such that pee) = 0.7 and p(F) = 0.5...
 6.2.6.2.12: Suppose that E and F are events such that pee) = 0.8 and p(F) = 0.6...
 6.2.6.2.13: Show that if E and F are events, then peE n F) 0:: pee) + p(F)  1....
 6.2.6.2.14: Use mathematical induction to prove the following generalization of...
 6.2.6.2.15: Show that if E I , E2, ... , En are events from a finite sample spa...
 6.2.6.2.16: Show that if E and F are independent events, then E and F are also ...
 6.2.6.2.17: If E and F are independent events, prove or disprove that E and F a...
 6.2.6.2.18: a) What is the probability that two people chosen at random were bo...
 6.2.6.2.19: a) What is the probability that two people chosen at random were bo...
 6.2.6.2.20: Find the smallest number of people you need to choose at random so ...
 6.2.6.2.21: Find the smallest number of people you need to choose at random so ...
 6.2.6.2.22: February 29 occurs only in leap years. Years divisible by 4, but no...
 6.2.6.2.23: What is the conditional probability that exactly four heads appear ...
 6.2.6.2.24: What is the conditional probability that exactly four heads appear ...
 6.2.6.2.25: What is the conditional probability that a randomly generated bit s...
 6.2.6.2.26: Let E be the event that a randomly generated bit string of length t...
 6.2.6.2.27: Let E and F be the events that a family ofn children has children o...
 6.2.6.2.28: Assume that the probability a child is a boy is 0.51 and that the s...
 6.2.6.2.29: A group of six people play the game of "odd person out" to determin...
 6.2.6.2.30: Find the probability that a randomly generated bit string of length...
 6.2.6.2.31: Find the probability that a family with five children does not have...
 6.2.6.2.32: Find the probability that a randomly generated bit string oflength ...
 6.2.6.2.33: Find the probability that the first child of a family with five chi...
 6.2.6.2.34: Find each of the following probabilities when n independent Bernoul...
 6.2.6.2.35: Find each of the following probabilities when n independent Bernoul...
 6.2.6.2.36: Use mathematical induction to prove that if E I, E2, ... , En is a ...
 6.2.6.2.37: (Requires calculus) Show that if E I, E2, is an infinite sequence o...
 6.2.6.2.38: A pair of dice is rolled in a remote location and when you ask an h...
 6.2.6.2.39: This exercise employs the probabilistic method to prove a result ab...
 6.2.6.2.40: Devise a Monte Carlo algorithm that determines whether a permutatio...
 6.2.6.2.41: Use pseudocode to write out the probabilistic primality test descri...
Solutions for Chapter 6.2: Discrete Probability
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 6.2: Discrete Probability
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 6.2: Discrete Probability have been answered, more than 36035 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Chapter 6.2: Discrete Probability includes 41 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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(DÂ» O.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.