- 12-3.1: P(2 vowels)
- 12-3.2: P(2 consonants
- 12-3.3: P(1 vowel, 1 consonant)
- 12-3.4: P(4 blue, 3 red, 3 yellow, in that order)
- 12-3.5: P(first 2 blue, last 2 blue)
- 12-3.6: P(H = 0)
- 12-3.7: P(H = 2)
- 12-3.8: P(3 football)
- 12-3.9: P(3 baseball)
- 12-3.10: P(1 basketball, 2 football)
- 12-3.11: P(2 basketball, 1 baseball)
- 12-3.12: P(1 football, 2 baseball)
- 12-3.13: P(1 basketball, 1 football, 1 baseball)
- 12-3.14: P(2 baseball, 2 basketball)
- 12-3.15: P(2 football, 1 hockey)
- 12-3.16: P(season 5 in the correct position)
- 12-3.17: P(seasons 1 and 8 in the correct positions)
- 12-3.18: P(seasons 1 through 4 in the correct positions)
- 12-3.19: P(all even-numbered seasons followed by all odd-numbered seasons)
- 12-3.20: P(all even-numbered seasons in the correct position)
- 12-3.21: P(seasons 5 through 8 in any order followed by seasons 1 through 4 ...
- 12-3.22: P(0 sophomores)
- 12-3.23: P(1 sophomore)
- 12-3.24: P(2 sophomores)
- 12-3.25: P(3 sophomores)
- 12-3.26: P(2 juniors)
- 12-3.27: P(1 junior)
- 12-3.28: LOTTERIES The state of Texas has a lottery in which 5 numbers out o...
- 12-3.29: P(math or statistics)
- 12-3.30: P(biological sciences)
- 12-3.31: P(physical sciences)
- 12-3.32: CARD GAMES The game of euchre (YOO ker) is played using only the 9s...
- 12-3.33: WRITING Josh types the five entries in the bibliography of his term...
- 12-3.34: OPEN ENDED Describe an event that has a probability of 0 and an eve...
- 12-3.35: Two dice are rolled. What is the probability that the sum will be 12?
- 12-3.36: A baseball player has 126 hits in 410 at-bats this season. What is ...
- 12-3.37: A hand of 2 cards is dealt from a standard deck of cards. What is t...
- 12-3.38: Writing in Math Use the information on page 697 to explain what pro...
- 12-3.39: ACT/SAT What is the value of _6! 2!? A 3 B 60 C 360 D 720
- 12-3.40: REVIEW A jar contains 4 red marbles, 3 green marbles, and 2 blue ma...
- 12-3.41: arranging 5 different books on a shelf
- 12-3.42: arranging the letters of the word arrange
- 12-3.43: picking 3 apples from the last 7 remaining at the grocery store
- 12-3.44: How many ways can 4 different gifts be placed into 4 different gift...
- 12-3.45: y x
- 12-3.46: y x
- 12-3.47: ab
- 12-3.48: bc
- 12-3.49: cd
- 12-3.50: bd
- 12-3.51: ac
Solutions for Chapter 12-3: Probability
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
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.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.