Chapter 9 Review
description
Transcript of Chapter 9 Review
Pre-Calculus
1/31/2007
Chapter 9 ReviewChapter 9 Review
an = a1 + (n – 1)d an = a1 • r (n – 1)
1aS
1 r
1 na an
2 1
n2a (n 1)d
2
n
kk 1
a
n1a (1 r )
1 r
n
kk 1
a
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Probability Review(Sections 9.1 – 9.3)
Probability Review(Sections 9.1 – 9.3)
Pre-Calculus
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Fundamental Principle of Counting(Multiplication Principle)
Fundamental Principle of Counting(Multiplication Principle)
n! = n(n – 1)(n – 2)(n – 3)… (2)(1)
A, B, C, D, E (5)(4)(3)(2)(1) = 120
How many ways can you arrange:
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PermutationsOrder is important!!!
PermutationsOrder is important!!!
permutations of “n” objects taken “r” at a time:
using “n” objects to fill “r” blanks in order.
How many ways can 6 runners finish 1 – 2 – 3?
nPr =n!
(n – r)!
6P3 =6!
(6 – 3)!720
6120= =
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only interested in the ways to select the “r” objects regardless of the order in which we arrange them.
How many ways can 2 cards be picked from a deck of 10:
nCr =n!
(n – r)! • r!
10C2 = = 4510!
(8)! • 2!= 3,628,800
40320 • 2
CombinationsOrder is NOT important!!!
CombinationsOrder is NOT important!!!
n
r
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Subsets of an “n” – setSubsets of an “n” – set
There are ________ subsets of a set with n objects (including the empty set and the entire set).
2n
DiMaggio’s Pizzeria offers patrons any combination of up to 10 different pizza toppings. How many different pizzas can be ordered if we can choose any number of toppings (0 through 10)?We could add up all the numbers of the form for r = 0, 1, …, 10 but there is an easier way. In considering each option of a topping, we have 2 choices: __________ or __________.
Therefore the number of different possible pizzas is:
Example:
Yes No
2n = 210 = 1024
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Binomial TheoremBinomial Theorem
n n 1 n r r nn n n n
(a b) a a b ... a b ... b0 1 r n
n r
nC
r Don’t forget:
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4 4 8 2
or 0.15452 52 52 13
Also, recall that in mathematics, the word or signifies addition; the word and signifies multiplication .
Also, recall that in mathematics, the word or signifies addition; the word and signifies multiplication .
Find the probability of selecting an ace or a king from a draw of one card from a standard deck of cards.
Find the probability of selecting an ace and a king from a draw of one card from a standard deck of cards.
4 4 16 1
or 0.005952 52 2704 169
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Venn Diagrams
sample space (all students)
subsets to represent “girls” and “sports”
1
(54%) 0.183
“boys”
“no sports”
decimals
1
students
girls sports
0.36 0.18 0.23
0.23
.54 .18 .36
1 .36 .18 .23 .23
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0.5
conditional probability
0.25 1
0.125 + 0.125 + 0.5 = 0.75
dependent
of the event A, given that event B occurs
2/4 or 0.5 1
along the branches that come out of the two jars
the probability P(A B)
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P(A andB)
P(A)
the ends of the branches
conditional probability formula
P(jar A and )
P(chocolate chip)
chocolate chip
P(A) • P(B A)
1 20.25 12 4
0.75 0.75 3
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1
6
binomial distribution
binomial Theorem
41 1 1 1 1
0.000776 6 6 6 65
6
45
0.482256
2 21 5
0.019296 6
2 21 5
6 6
2 2
4 2
1 5C 0.11574
6 6
4C2 = 6 6
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Homework AnswersHomework Answers# 54, 56, 61, 65, 68, 70, 76, 77 , 17, 23
(p. 748 – 749)
5
5 4 3 2 2 3 4 5
5 4 3 2 2 3 4 5
5 4 3 2 2 3 4 5
(2x y)
5 5 5 5 5 5(2x) (2x) (y) (2x) (y) (2x) (y) (2x)(y) (y)
0 1 2 3 4 5
(1)32x (5)16x y (10)8x y (10)4x y (5)2xy (1)y
32x 80x y 80x y 40x y 10xy y
11
8 3 8 8
(x 2)
11x ( 2) 165 ( 8)x 1320x
3
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Probability ReviewQuestions
Probability ReviewQuestions
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Probability ExerciseProbability Exercise
Suppose there is a 70% chance of rain tomorrow. If it rains, there is a 10% chance that all of the rides at an amusement park will be operating. If it doesn’t rain, there is a 95% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?
.7(.1) .3(.95) .355
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Probability ExerciseProbability Exercise
Binomial Theorem:
Bubba rolls a fair die 6 times. What is the probability that he will roll exactly two 2’s?
2 4
6 2
1 5C 0.2009
6 6
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Probability ExerciseProbability Exercise
There are 20 runners on a track team.
How many groups of 4 can be selected to run the 4 x 100 relay?
How many ways can 4 runners be selected to run 1st – 2nd – 3rd – 4th?
20 4C 4845
20 4P 116280
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Probability ExerciseProbability Exercise
A fair coin is tossed 10 times.
Find the probability of tossing HHHHHTTTTT.
Find the probability of tossing exactly 5 tails in those 10 tosses
10
10 5
1 252C .2461
2 1024
101 1
2 1024
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Review QuestionReview Question
# 43
(p. 748)
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Probability ExerciseProbability Exercise
Find the x4 term in the expansion of:
5(4x y)
4 1 45 1C (4x) (y) 1280x y
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Probability ExerciseProbability Exercise
License plates are created using 3 letters of the alphabet for the first 3 characters and 4 numbers for the last 4 characters.
How many possible different license plates are there if the letters and numbers are NOT allowed to repeat?
26 25 24 10 9 8 7 78,624,000
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Probability ExerciseProbability Exercise
A spinner, numbered 1 through 10, is spun twice.
What is the probability of spinning a 1 and a 10 in any order?
What is the probability of not spinning the same number twice?
2 1 1
10 10 50
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Probability ExerciseProbability Exercise
Expand: 4(3x 2y)
4 0 3 1 2 24 0 4 1 4 2
1 3 0 44 3 4 4
C (3x) (2y) C (3x) (2y) C (3x) (2y)
C (3x) (2y) C (3x) (2y)
4 3 2 2 3 4(3x) 4(3x) (2y) 6(3x) (2y) 4(3x)(2y) (2y)
4 3 2 2 3 481x 216x y 216x y 96xy 16y
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an = 12 – 2.5(n – 1)an = an-1 – 2.5
Review QuestionReview Question
Is the series arithmetic or geometric?Find the explicit formulaFind the recursive formulaFind the 100th termFind the sum for a1 through a100
12, 9.5, 7, 4.5, …
10, 12, 14.4, 17.28, …
12 – 2.5(100 – 1) = – 235.5
12 235.5
100 1117.52
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an = 10(1.2)(n – 1)
Review QuestionReview Question
Is the series arithmetic or geometric?Find the explicit formulaFind the recursive formulaFind the 100th termFind the sum for a1 through a100
12, 9.5, 7, 4.5, …
10, 12, 14.4, 17.28, …
an = an-1(2.5)
10(1.2)99 = 690,149,787.7
10010(1 1.2 )4,140,898,676
1 1.2
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Review QuestionReview Question
Evaluate:
6
212
k 3
( k )
– ½(3)2 – ½(4)2 – ½(5)2 – ½(6)2
– ½(9) – ½(16) – ½(25) – ½(36)
– 43
Pre-Calculus
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Review QuestionReview Question
Is this sequence arithmetic or geometric?
9, 18, … 144