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WARM ­ UP
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9.1 TOPICS
• Factorial Review
• Counting Principle
• Permutations
• Distinguishable permutations
• Combinations
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FACTORIAL REVIEW
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Question...
How many sandwiches can you make if you have 3 types of
bread, 4 types of meat and 2 types of cheese?
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Same Question...
How many sandwiches can you make if you have 3 types of
bread, 4 types of meat and 2 types of cheese?
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EXAMPLE #1
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Permutation ­ ways that a set of objects can be arranged
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EXAMPLE #2
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How many distinguishable permutations (unique arrangements) of the
words below can you make?
MOP
MOM
MOP
MPO
OMP
OPM
POM
PMO
MOM
MMO
OMM
OM M
MOM
MMO
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How many distinguishable permutations (unique arrangements) of the
word below can you make?
CANADA
CA NA D A
CA NA D A
CA NA D A
CA NA D A
CA NA D A
CA NA D A
and more..... 11
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DISTINGUISHABLE PERMUTATIONS
Let S be a set of n elements of k different types. Let n1 = the number of elements of type 1, n2 = the number of elements of type 2 ......nk =the number of elements of type k. Then the number of distinguishable permutations of the n elements is:
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EXAMPLE #3
Find the number of distinguishable 13­letter "words" that can be made from the letters in MASSACHUSETTS?
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Permutations
You can also make the blanks and fill them in. It accomplishes the same result.
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EXAMPLE #4
Your band has made 12 songs and plans to record 9 of them for a CD. In how many ways can you arrange the songs for the CD?
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COMBINATIONS
You also can still use the fill in the blank approach, you just need to make sure that you divide by the number of things that are the same.
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EXAMPLE #5
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Can you tell the difference?
Question...
Out of 20 seniors, 3 are chosen to get 3 scholarships. One is worth $1500, one is worth $1000, one is worth $500. How many ways are there to do this?
Permutation or Combination?
Do you care which scholarship you get???
This is called finding a permutation. The order in which each thing is picked is important. It matters if you are picked first or last.
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Can you tell the difference?
Question... does order matter?
Out of 20 seniors, 3 are chosen to get 3 ­ $500 scholarships. How many ways are there to do this? Permutation or Combination?
Combination
But, now there is more that you have to do. Unlike the first problem, the order in which you get picked doesn't matter. It doesn't matter if you get the 1st scholarship or the third one, you are still getting $500
This is called finding a combination. A combination is basically a permutation where it doesn't matter the order in which people were picked
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Thinking of the same question...
Suppose Tom, Mary and John win. There are 6 different ways that they can be picked:
TMJ, TJM, MJT, MTJ, JMT, JTM
All 6 ways have the same people winning the same scholarships, so this is the combinations of how they can be picked.
So the total from before needs to be divided by 6 or (3!)
That's the difference between a permutation and a combination.
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The short and sweet version...
Use Permutations when the order of selection matters
Use Combinations when the order of selection does not matter
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Practice Time
1) A company advertises two job openings, one for copy writer and one for artist. If 10 people who are qualified for either position apply, in how many ways can the opening be filled?
2) A company advertises two job openings for computer programmers, both with the same salary and job description. In how many ways can the openings be filled if 10 people apply?
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Assignment 9.1 p.649
#1­12, 19­26, 29, 31, 33, 34
36­39, 41, 42, 45, 46, 48
(This is the assignment for Monday
and Tuesday)
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Out of 20 seniors, 3 are chosen to get 3 ­ $500 scholarships. How many ways are there to do this? So, the solution is:
P
I
K
S
This is called finding a combination. A combination is basically a permutation where it doesn't matter the order in which people were picked
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COMBINATIONS
P
I
K
S
How many permutations would there be?
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P
I
K
S
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PERMUTATIONS
P
I
K
S
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P
I
K
S
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QUESTION
Out of 20 seniors, 3 are chosen to get 3 scholarships. One is worth $1500, one is worth $1000, one is worth $500. How many ways are there to do this?
P
I
K
S
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Same question
Out of 20 seniors, 3 are chosen to get 3 ­ $500 scholarships. How many ways are there to do this? Start out like before doing a permutation:
P
I
K
S
But, now there is more that you have to do. Unlike the first problem, the order in which you get picked doesn't matter. It doesn't matter if you get the 1st scholarship or the third one, you are still getting $500
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P
I
K
S
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P
I
K
S
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Review from yesterday
1) Suppose a club with 12 members wishes to choose a president, a vice­president and a treasurer. In how many ways can this be done?
2) Suppose on the other hand that the club merely wants to choose a governing council of three. In how many ways can this be done
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9.2 day1 TOPICS
• P and C backwards
• BINOMIAL Expansion
• Pascal's Triangle
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Example #1 (REVIEW)
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Now what...think about the set up. You WILL SEE THIS TYPE OF QUESTION AGAIN!!!
Example #2
How many permutations are possible when seating 4 people around a circular table?
* We need to choose where the first person sits... then from there we have 3 places to full so.... 36
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CARDS
Example #3
How many ways are there to deal 13 cards from a standard deck of cards if the order in which the cards are dealt is (a) important? (b) not important?
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BINOMIAL EXPANSION
Expand the following:
(x + y)0 1
(x + y)1 x + y
(x + y)2 x2 + 2xy +y2
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Expand the following:
(x + y)3 (x + y)(x + y)2 (x + y)(x2 + 2xy + y2 )
x3 + 3x2y + 3xy2 + y3 39
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Expand the following:
(x + y)4 (x + y)2 (x + y)2
(x2 + 2xy + y2 )(x2 + 2xy + y2 )
x4 + 4x3y + 6x2y2 + 4xy3 + y4
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PATTERNS...what will the next line be?
(x + y)0 1
(x + y)1 x + y
(x + y)2 x2 + 2xy +y2
(x + y)3 x3 + 3x2y + 3xy2 + y3 (x + y)4 x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x + y)5
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(x + y)0 (x + y)1 (x + y)2 May 06, 2014
1
1 1
1
1 2 1
x + y
1 3 3 1
2
x2 + 2xy +y 1 4 6 4 1
(x + y)3 x3 + 3x2y + 3xy2 + y3 (x + y)4 x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x + y)5
Pascal's Triangle
One use is to find the coefficients of an expansion of (x + y)n
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nCr
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=
=
= Pascal's Triangle
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We can use nCr to find coefficients...
Example #4
Find the first 5 terms of (x + y)20
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Here's How...
Find the first 5 terms of (x + y)20
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On the calculator...
But what if I want all the numbers in a certain row from Pascal's Triangle so I can expand a binomial?
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So Make a Table...
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Assignment 9.2 p.656
#1­15 odd
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9.2 day2 TOPICS
• Expansions and terms
• Word problem and expansion
• TABLE CHALLENGE
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EXAMPLE #1
Find the fourth term of (x ­ 2y)10
***Do more than one term to show alternating signs when expanded versus all "+" with (x + y) versus (x ­ y)
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EXAMPLE #2
Find the a4 term of (2a ­ b2 )6
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TABLE CHALLENGE
1) There are 14 people in a club. A committee of 6 persons are to be chosen to
represent the club at a conference. In how many ways can the committee be chosen?
2) Suppose the letters in VERMONT are used to form "words". How many "words" can be formed that begin with a vowel and end with a consonant?
3) Suppose a True-False Test has 25 questions.
a) In how many ways may a student mark the test if all questions are answered?
b) In how many ways may a student mark the test if questions could be left blank?
c) In how many ways may a student mark a test if 11 questions are answered correctly, 14 are incorrect,
provided all questions are answered?
4) Expand: (x+y)9
5) In how many ways may 9 people be seated around a circular table?
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Remember...
nCr
=
=
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Assignment 9.2 p.656
#17­29 odd, 34­40 all
WORKSHEET ­­ due next Thursday
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Now for Binomial Expansion
6
Expand: (x+y)
( x + y ) 6
P
I
K
S
( x + y ) 6
( x + y ) 6
( x + y ) 6
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(a + b)n = How do we use this to answer questions?
Example #3
Find how many ways you can hire 20 females and 15 males.
(M + F)35 : 35C15 M15F
20
P
I
K
S
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I
K
S
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PRACTICE for QUIZ
1)
2)
3)
P
I
7
K
S
9
4) Expand: (x + y)
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P
I
K
S
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Prove for all integers n≥2
P
I
K
S
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Now go backwards...
Example #1
Solve for n: n C 2 = 45
P
I
Solve for n: n C 2 = n­1 P 2
K
S
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9.3 day1 TOPICS
• Probability
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Testing Your Intuition About Probability
Find the probability of each of the following events.
1.
Tossing a head on one toss of a fair coin
2.
Tossing two heads in a row on two tosses of a fair coin
3.
Drawing a queen from a standard deck of 52 cards
4.
Rolling a sum of 4 on a single roll of two fair dice
5.
Guessing all 6 numbers in a state lottery that requires you to pick 6
numbers between 1 and 46 inclusive
1/8,099,197,920
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Probability of an Event (Equally Likely Outcomes)
If E is an event in a finite, nonempty sample space S of equally likely
outcomes, then the probablility of the event E is
Probability Distribution
Each outcome is assigned a unique probability
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Probability of an Event (Equally Likely Outcomes)
Find the probability of rolling a sum divisible by 3 on a single roll of two
fair dice
Make a sample space for sums found by rolling 2 fair dice
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
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Probability Function (Distribution)
A probability function is a function P that assigns a real number to each
outcome in a sample space S subject to the following conditions:
1. 0 ≤ P(O) ≤ 1 for every outcome O
2. the sum of the probability of all outcomes in S is 1
3. P(∅) = 0
EXAMPLE 1
Find the probability of rolling a sum divisible by 3 on a single roll of 2 fair dice.
First find the probability distribution. Then add all your probabilities together from your favorable outcomes.
Outcome
2 3 4 5 6
7 8 9 10 11 12
Probability
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EXAMPLE 2 - Testing a Probability Function
Is it possible to weight a standard 6 - sided die in such a way that the
probability of rolling each number n is exactly 1/(n2 + 1)?
Hint: What would the probability distribution look like?
The probability distribution would look like this:
Outcomes
Probabilities
1
2
3
4
5
6
1/2
1/5
1/10
1/17
1/26
1/37
This is not a valid probability function because the probabilities do not
equal 1
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Probability of an Event (Outcomes Not Equally Likely)
Let S be a finite, nonempty space in which every outcome has a probability
assigned to it by a probability function P. If E is any event in S, the
probability of the event E is the sum of the probabilities of all outcomes
contained in E.
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Strategy for Determining Probabilities
1.
Determine the sample space of all possible outcomes. When possible
choose outcomes that are equally likely
2.
If the sample space has equally likely outcomes, the probability of an
event E is determined by counting:
3.
If the sample space does not have equally likely outcomes, determine
the probability function. Check to be sure that the conditions of a
probability function are satisfied. Then the probability of an event E
is determined by adding up the probabilities of all the outcomes
contained in E
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Multiple Principle of Probability
Suppose an event A has probability p1 and an event B has probability p2
under the assumption that A occurs. Then the probability that both A and B
occur is p1p2
EXAMPLE 3
Sal opens a box of a dozen cremes and generously offers two of them to Val.
Val likes vanilla cremes the best, but all the chocolates look alike on the
outside. If four of the twelve cremes are vanilla, what is the probability
that both of Val's picks turn out to be vanilla?
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Probability using Combinations (Instead of Multiplication Principle)
EXAMPLE 3: Same question, different method:
Sal opens a box of a dozen cremes and generously offers two of them
to Val. Val likes vanilla cremes the best, but all the chocolates look
alike on the outside. If four of the twelve cremes are vanilla, what is
the probability that both of Val's picks turn out to be vanilla?
Remember: From the four vanilla cremes, Val wants 2 so
4C2
=6
From the box of 12, Val chooses 2 so 12C2 = 66
Therefore P(E) = 6/66 or 1/11
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Venn Diagrams
EXAMPLE 4:
In a large high school, 54% of the students are girls and 62% of the
students play sports. Half of the girls play sports.
a)
What percentage of the students who play sports are boys?
b)
If a student is chosen at random, what is the probability that it is a
boy who does not play sports?
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Tree Diagrams
EXAMPLE 5
Two identical cookie jars are on a counter. Jar A contains 2 chocolate
chip cookies and 2 peanut butter cookies, while Jar B contains 1 chocolate
chip cookie. We select a cookie at random. What is the probability that
it is a chocolate chip cookie?
0.5
0.5
Jar
A
Jar
B
CC
0.25
0.25 CC
0.25
PB
0.25
PB
1
P(choc chip) = CC
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Conditional Probability Formula
If the event B depends on the event A, then
EXAMPLE 6
Two identical cookie jars are on a counter. Jar X contains 2 chocolate
chip and 2 peanut butter cookies, while jar Y contains 1 chocolate chip
cookie.
Suppose you draw a cookie at random from one of the jars. Given that it
is a chocolate chip cookie, what is the probability it came from jar X?
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EXAMPLE 6 ­ Solution
Two identical cookie jars are on a counter. Jar X contains 2 chocolate
chip and 2 peanut butter cookies, while jar Y contains 1 chocolate chip
cookie.
Suppose you draw a cookie at random from one of the jars. Given that it
is a chocolate chip cookie, what is the probability it came from jar X?
0.5
Jar
X
CC
0.25
0.25 CC
0.25
PB
0.25
0.5
Jar
Y
P( chocolate chip)
1
PB
CC
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Assignment 9.3 p.666
#1­27 odd
WORKSHEET Permutation and Combinations­­ due Thursday
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9.3 day2 TOPICS
• Probability and Binomial
Expansions
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Remember Binomial Expansion from last week?
(a + b)n = How do we use this to answer questions?
With questions such as:
Find how many ways you can hire 20 females and 15 males.
(M + F)35 : 35C15 M15F20
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Binomial Distributions (REALLY IMPORTANT NOTES)
For events that have 2 possible outcomes (example: heads or tails) we can
use the Binomial theorem that we learned last section to determine
probabilities.
(X + Y)
n
number of trials
Probability of each outcome, ALWAYS sum to 1.
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EXAMPLE 1
Suppose you roll a die 4 times. Find the probability that you roll exactly two 3's.
Let x = probability of rolling a 3 (which is what??)
Let y = probability of not rolling a 3 (which is what??)
The probability that you need can be found by looking at the
binomial expansion of (x + y)4
Remember that (x + y)4 = 4C0x4 + 4C1x3y + 4C2x2y2 + 4C3xy3 + 4C4y4
= x4+ 4x3y + 6x2y2 +4xy3 + y4
We are looking for the term x2y2 (2 3's and 2 not 3's)
4C 2x
2 2
y
P(exactly 2 3's) = 6(1/6)2(5/6)2
= 0.11574
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EXAMPLE 2
Suppose you roll a die 4 times.
What is the probability of rolling all 3's?
EXAMPLE 3
Suppose you roll a die 4 times.
What is the probability of rolling at least 2 3's?
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EXAMPLE 4 (A,B, and C)
Suppose Michael makes 90% of his free throws. If he shoots 20 free
throws, and if his chances of making each one is independent of the other
shots, what is the probability he makes:
A) All 20
B)
Exactly 18
C)
At least 18
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EXAMPLE 4 (A,B, and C)
Suppose Michael makes 90% of his free throws. If he shoots 20 free
throws, and if his chances of making each one is independent of the other
shots, what is the probability he makes:
a) All 20
b)
Exactly 18
c)
At least 18
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Assignment 9.3 p.666
#28, 31, 33, 35, 39, 41
53­56
Add today's examples into your Stenos AND expand (2x ­ 3y)4
WORKSHEET Permutation and Combinations­­ due Thursday
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Probability
outcome: a possible result of an experiment
event: is an outcome
sample space: set of ALL possible outcomes
What is the sample space of flipping a coin and rolling a die?
Probability of an event⇒
As a % : 0 - 100%
As a # : 0 - 1
25% - unlikely
75% - likely
P(event)
The measure of the likelihood, or chance, that the
event WILL occur. Can be expressed as a decimal,
fraction, or percent.
50% - equally likely to happen or not happen
100% - certain
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2 Types of probability
Theoretical Probability =
what SHOULD
happen
# of favorable outcomes
Total # of outcomes
Experimental Probability =
# of successes
# of trials
what happened
after actually
doing the
experiment
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What is the Theoretical Probability that the spinner lands on blue?
P(blue) =
What is the Experimental Probability that the spinner lands on blue?
P(blue) =
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You need Combination formula
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9.4 TOPICS
• Arithmetic Sequences
• Geometric Sequences
• Explicit
• Recursive
• Diverge vs Converge
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Notations you need to understand:
n
a1
an
an­1
Recursive Formula: Formula based on knowing the previous term.
NOW = NEXT
EXPLICIT Formula: Formula used to find the VALUE of a term knowing which term you are testing.
Find the nth term
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explicit
starting term
How do you find the common difference?
value of that term
nth term
difference between the values
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Practice
1. Find the first 3 terms in the sequence where a1=6 and d=2.5
2. Find the nth term of the following sequence: a1 = 5, d= 1/2, n=13
3. 203 is the _________th term of ­5, ­1, 3...
4. Find the arithmetic means in the sequence: 21, ___, ___, ­18, ___ 100
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explicit formula is
How do you calculate the ratio?
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Practice
1. Find the first 5 terms in the sequence where a1=4 and r=1/2
2. Find the nth term of the following sequence: a1 = 5, r= ­3, n=7
3. 62,500 is the _________th term of 4, 20, 100...
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Arithmetic, Geometric or Neither
3, 8, 13, 18.......
4, 8, 16, 32......
1, ­3, ­5, ­7,........
27, ­18, 12, ­8,...........
2, 5, 7, 10.......
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Explicit or Recursive
Explicit­ tn is given in terms of n. Do not need to know any of the previous terms to find a specific term
Arithmetic Explicit
Geometric Explicit
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Recursive­ a formula that gives a rule for tn in terms of tn­1
Recursive definition consists of two parts:
An initial condition that tells where the sequence starts
An equation that tells how any term in the sequence is related to the preceding term
For each of the recursive formulas given below, given the first 4 terms of the sequence and write an explicit formula for the sequence if possible
t1 = 3
tn = 2tn­1 + 1
t1 = 23
tn = tn­1 ­ 3
t1 = 64
tn = .5tn­1
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EXAMPLE #1
The fourth and seventh terms of an arithmetic sequence are ­8 and 4, respectively. Find the first term and a recursive rule for the nth term.
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EXAMPLE #2
The second and eighth terms of a geometric sequence are 3 and 192, respectively. Find the first term, the common ratio and the explicit rule for the nth term.
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EXAMPLE #3
The population of a city is growing at a rate of 15% each year. How long before the city triples in size?
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THOUGHT
Is the sequence 2, 2, 2, 2, ......
Geometric, Arithmetic, Neither or Both?
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Assignment 9.4 p.676
#1­20, 21­33 mod 3
37, 39, 43­48
WORKSHEET ­­ due TOMORROW
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Practice Questions for Quiz
1.
If it rains tomorrow, the probability that John will practice piano is 0.8. If
it doesn't rain, there is only a 0.4 chance that John will practice. Suppose
the chance of rain is 60% tomorrow. What is the probability that john will
practice?
2.
Out of forty students, 14 are taking English Composition and 29 are taking
Chemistry. If five students are in both classes, how many students are in
neither class? How many are in either class?
3.
You toss a coin 15 times. What is the probability you get 9 heads and 6
tails?
4.
A jar has 5 blue marbles, 7 red marbles and 9 yellow marbles. What is the
probability of drawing a blue and then a yellow if (a) the first marble is not
replaced before drawing the second, (b) the first is put back before drawing
the second?
5.
Find the first 8 terms of the recursively defined sequence:
t1 = 5, tn = 3(tn-1) - 2
6.
Find the first 4 terms of the explicitly defined sequence:
t n = n2 + 1
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Ch 9 SB answers.notebook
May 06, 2014
Practice Questions for Quiz
1.
If it rains tomorrow, the probability that John will practice piano is 0.8. If
it doesn't rain, there is only a 0.4 chance that John will practice. Suppose
the chance of rain is 60% tomorrow. What is the probability that john will
practice?
2.
Out of forty students, 14 are taking English Composition and 29 are taking
Chemistry. If five students are in both classes, how many students are in
neither class? How many are in either class?
3.
You toss a coin 15 times. What is the probability you get 9 heads and 6
tails?
4.
A jar has 5 blue marbles, 7 red marbles and 9 yellow marbles. What is the
probability of drawing a blue and then a yellow if (a) the first marble is not
replaced before drawing the second, (b) the first is put back before drawing
the second?
5.
Find the first 8 terms of the recursively defined sequence:
t1 = 5, tn = 3(tn-1) - 2
6.
Find the first 4 terms of the explicitly defined sequence:
tn = n2 + 1
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Ch 9 SB answers.notebook
May 06, 2014
Practice Questions for Quiz
1.
If the school cafeteria serves meat loaf, there is a 70% chance that they
will serve peas. If they do not serve meat loaf, there is a 30% chance they
will serve peas anyway. Meat loaf will be served exactly once during the 5day week. If you are going to eat lunch today, what is the probability that
peas are on the menu?
2.
A survey of 1000 people was conducted by a local ice cream shop. Two
questions were asked: Do you like chocolate? Do you like vanilla?
The probability of a person liking chocolate is 45%. The probability of a
person liking vanilla is 52%. 50% of the people who like vanilla also like
chocolate. How many people do not like either chocolate or vanilla?
3.
You toss a coin 18 times. What is the probability you get 12 heads and 6
tails?
4.
If the track is wet, Champion has a 70% chance of winning the fifth race at
Belmont. If the track is dry, she only has a 40% chance of winning.
Weather forecasts an 80% chance the track will be wet. Find the
probability that Chamion wins the race.
5.
Find the first 4 terms of the recursively defined sequence:
t1 = -1, tn = (tn-1)2 + 2
6.
Find the first 4 terms of the explicitly defined sequence:
tn = 4n + 3
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Ch 9 SB answers.notebook
May 06, 2014
Practice Questions for Quiz
1.
If the school cafeteria serves meat loaf, there is a 70% chance that they
will serve peas. If they do not serve meat loaf, there is a 30% chance they
will serve peas anyway. Meat loaf will be served exactly once during the 5day week. If you are going to eat lunch today, what is the probability that
peas are on the menu?
2.
A survey of 1000 people was conducted by a local ice cream shop. Two
questions were asked: Do you like chocolate? Do you like vanilla?
The probability of a person liking chocolate is 45%. The probability of a
person liking vanilla is 52%. 50% of the people who like vanilla also like
chocolate. How many people do not like either chocolate or vanilla?
3.
You toss a coin 18 times. What is the probability you get 12 heads and 6
tails?
4.
If the track is wet, Champion has a 70% chance of winning the fifth race at
Belmont. If the track is dry, she only has a 40% chance of winning.
Weather forecasts an 80% chance the track will be wet. Find the
probability that Chamion wins the race.
5.
Find the first 4 terms of the recursively defined sequence:
t1 = -1, tn = (tn-1)2 + 2
6.
Find the first 4 terms of the explicitly defined sequence:
tn = 4n + 3
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May 06, 2014
EXAMPLES
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May 06, 2014
EXAMPLE
The second and eighth terms of a geometric sequence are 3 and 192, respectively. Find the first term, the common ratio and the explicit rule for the nth term.
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May 06, 2014
EXAMPLE
The fourth and seventh terms of an arithmetic sequence are ­8 and 4, respectively. Find the first term and a recursive rule for the nth term.
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May 06, 2014
EXAMPLE
The population of a certain country grows as a result of two conditions:
1) The annual population growth is 1% of those already in the country
2) 20,000 people immigrate into the country each year
If the population is now 5,000,000 people, what will the population be in two years?
In 20 years?
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May 06, 2014
9.5 day1 TOPICS
• Review of 9.4
• Finite Series
• Sums of Finite Series
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May 06, 2014
EXAMPLE #1 (review)
The third and 8th terms of an arithmetic sequence are 1 and 11, respectively. Find the first term, the common difference, and a explicit rule for the nth term.
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May 06, 2014
EXAMPLE #2­3 (REVIEW)
#2.
#3.
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May 06, 2014
EXAMPLE #2­3 (REVIEW)
#2.
#3.
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May 06, 2014
Summation Notation
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Example #4
Find the SUM of each:
A)
C)
B)
D)
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May 06, 2014
Example #5 Express each of the following series using sigma notation
A)
B)
3 + 6 + 9 + 12 + .....+ 300
1 + 8 + 27 + 64 + 125
C)
D)
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If you know:
n = # of terms
a1 = 1st term
an = Last term
May 06, 2014
If you know:
n = # of terms
a1 = 1st term
d = common difference
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May 06, 2014
EXAMPLES #6 and #7- Find the sum of the arithmetic sequences.
6. -7, -3, 1, 5, 9, 13
7. 117, 110, 103, ...., 33
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May 06, 2014
If you know:
n = # of terms
a1 = 1st term
r = ratio
EXAMPLES #8 and #9 - Find the sum of the geometric sequence.
8. 5, 15, 45, ... ,
98,145
9. 42, -7, 7/6, ...., 42(-1/6)9
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May 06, 2014
Assignment 9.5 p.684
#3­23, mod 3, 35, 36, 37
From 9.5 Day 1 Choose 1 from 1­3, 2 from Ex 4, 2 from Ex 5, 6, 8, 9
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Do you have any questions?
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9.5 day2 TOPICS
• Converge and Diverge (9.4)
• Infinite Series
• Sums of Infinite Series
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What would the sum of the series be if the series had infinite terms?
2 + 4 + 8 + 16 + 32................
.1 + .01 + .001 + .0001..............
The first series diverges because as you add more terms the sum gets bigger with no end The second series converges because eventually you will add such a small amount that the sum does not get bigger
How do you tell if a series converges or diverges?
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May 06, 2014
Geometric Series
IMPORTANT
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May 06, 2014
EXAMPLE #1
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May 06, 2014
EXAMPLES
Does the following series diverge or converge? If it converges, give its sum
2) 6 + 3 + 3/2 + 3/4 + ...
3) 1/48 + 1/16 + 3/16 + 9/16 + ...
4) 4 - 2 + 1 - 1/2 + 1/4 ...
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May 06, 2014
EXAMPLE #5-- WORD PROBLEM
A ball is dropped from a height of 5 feet. Each time it
rebounds to 80% of the last bounce. What is the total distance
that the ball travels before coming to rest?
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May 06, 2014
REPEATING DECIMALS
Express 0. 12121212.... as a fraction
First, write as a series:
.12 + .0012 + .000012 + ...
Then determine r
r = .01
Then find the "sum"
a
1-r
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May 06, 2014
REPEATING DECIMALS...again
Express 0. 12121212.... as a fraction
n = .1212121212....
100n = 12.12121212...
- n = 0.12121212......
99n = 12
n = 12/99
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May 06, 2014
EXAMPLE #6
Write .234234234.... as a fraction
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May 06, 2014
Assignment 9.5 p.684
#25­32, 39, 41­46
Stenoes: Today: Pick 2 from #1­4 and 5 & 6
From 9.5 Day 1(Yesterday) Choose 1 from 1­3, 2 from Ex 4, 2 from Ex 5, 6, 8, 9
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May 06, 2014
REVIEW and WARM ­ UP
State the first 4 terms of the sequence, then state if the sequence is arithmetic, geometric or neither. tn = 5n +2
tn = 3n
tn = n3
tn =
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May 06, 2014
Give each series in expanded form
k
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EXAMPLES - Find the sum of the first n terms. The sequence is
either arithmetic or geometric.
1. 2, 5, 8, .... ; n = 10
2. 4, -2, 1, -.5, ..., ; n = 12
3. 1, 2, 3, ..., ; n = 500
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May 06, 2014
Today's Plan
Probability Worksheet - Due TODAY!!!
Series and Sequence Worksheet - Due Friday
Notebook Assignments - Due Wednesday
Study for TEST!!! - material on web page
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May 06, 2014
Today's Plan
Series and Sequence Worksheet - Due Friday
Standard Prep 9 -- Due Wednesday
Notebook Assignments - Due Wednesday
Study for TEST!!! - material on web page
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May 06, 2014
Today's Plan
Series and Sequence Worksheet - Due TODAY!
TAKE HOME QUIZ - DUE MONDAY!!
Standard Prep 9 -- Due Wednesday
Notebook Assignments - Due Wednesday
Study for TEST!!! - material on web page
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Today's Plan
TAKE HOME QUIZ - DUE TODAY!!
Standard Prep 9 -- Due Wednesday
Notebook Assignments - Due Wednesday
Study for TEST!!! - material on web page
Projects Assigned tomorrow - find a group!
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May 06, 2014
Today's Plan
Standard Prep 9 -- Due Wednesday
Notebook Assignments - Due Wednesday
Study for TEST!!! - material on web page
Projects Assigned TODAY!!! Know your group!
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May 06, 2014
Converge vs Diverge
Let {an} be a sequence of real numbers and consider
If the limit is a finite number L, the sequences
converges and L is the limit of the sequences.
If the limit is infinite or nonexistent, the sequence
diverges
Examples:
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