 6.1.6.1.1: What is the probability that a card selected from a deck is an ace?
 6.1.6.1.2: What is the probability that a die comes up six when it is rolled?
 6.1.6.1.3: What is the probability that a randomly selected integer chosen fro...
 6.1.6.1.4: What is the probability that a randomly selected day of the year (f...
 6.1.6.1.5: What is the probability that the sum of the numbers on two dice is ...
 6.1.6.1.6: What is the probability that a card selected from a deck is an ace ...
 6.1.6.1.7: What is the probability that when a coin is flipped six times in a ...
 6.1.6.1.8: What is the probability that a fivecard poker hand contains the ac...
 6.1.6.1.9: What is the probability that a fivecard poker hand does not contai...
 6.1.6.1.10: What is the probability that a fivecard poker hand contains the tw...
 6.1.6.1.11: What is the probability that a fivecard poker hand contains the tw...
 6.1.6.1.12: What is the probability that a fivecard poker hand contains exactl...
 6.1.6.1.13: What is the probability that a fivecard poker hand contains at lea...
 6.1.6.1.14: What is the probability that a fivecard poker hand contains cards ...
 6.1.6.1.15: What is the probability that a fivecard poker hand contains two pa...
 6.1.6.1.16: What is the probability that a fivecard poker hand contains a flus...
 6.1.6.1.17: What is the probability that a fivecard poker hand contains a stra...
 6.1.6.1.18: What is the probability that a fivecard poker hand contains a stra...
 6.1.6.1.19: What is the probability that a fivecard poker hand contains cards ...
 6.1.6.1.20: What is the probability that a fivecard poker hand contains a roya...
 6.1.6.1.21: What is the probability that a die never comes up an even number wh...
 6.1.6.1.22: What is the probability that a positive integer not exceeding 100 s...
 6.1.6.1.23: What is the probability that a positive integer not exceeding 100 s...
 6.1.6.1.24: Find the probability of winning the lottery by selecting the correc...
 6.1.6.1.25: Find the probability of winning the lottery by selecting the correc...
 6.1.6.1.26: Find the probability of selecting none of the correct six integers,...
 6.1.6.1.27: Find the probability of selecting exactly one of the correct six in...
 6.1.6.1.28: In a superlottery, a player selects 7 numbers out of the first 80 p...
 6.1.6.1.29: In a superlottery, players win a fortune if they choose the eight n...
 6.1.6.1.30: What is the probability that a player wins the prize offered for co...
 6.1.6.1.31: Suppose that 100 people enter a contest and that different winners ...
 6.1.6.1.32: Suppose that 100 people enter a contest and that different winners ...
 6.1.6.1.33: What is the probability that Abby, Barry, and Sylvia win the first,...
 6.1.6.1.34: What is the probability that Bo, Colleen, Jeff, and Rohini win the ...
 6.1.6.1.35: In roulette, a wheel with 38 numbers is spun. Ofthese, 18 are red, ...
 6.1.6.1.36: Which is more likely: rolling a total of 8 when two dice are rolled...
 6.1.6.1.37: Which is more likely: rolling a total of 9 when two dice are rolled...
 6.1.6.1.38: Two events EI and E2 are called independent if p(EI n E2) = p(EJ )p...
 6.1.6.1.39: Explain what is wrong with the statement that in the Monty Hall Thr...
 6.1.6.1.40: Suppose that instead of three doors, there are four doors in the Mo...
 6.1.6.1.41: This problem was posed by the Chevalier de Mere and was solved by B...
Solutions for Chapter 6.1: Discrete Probability
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 6.1: Discrete Probability
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.1: Discrete Probability includes 41 full stepbystep solutions. Since 41 problems in chapter 6.1: Discrete Probability have been answered, more than 40165 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

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

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

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).

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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.

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·