10.1 10.1 Infinite Sequences and Summation Notation

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10.110.1 Infinite Sequences and
Summation Notation

• Find terms of sequences given the general or
  nth term.
• Convert between sigma notation and other notation
  for a series.

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Sequences

• A sequence is a function.
• An infinite sequence is a function 1, 2, 3, 4,
  5, .
• A finite sequence is a function 1, 2, 3, 4, 5,
  , n, for some positive integer n.

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Sequence Formulas

• In a formula, the function values are known as
  terms of the sequence.
• The first term in a sequence is denoted as a1,
• the third term as a3 ,
• and the nth term, or the general term, as an.

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Example

• Find the first 4 terms and the 9th term of the
  sequence whose general term is given by an
  4(?2)n.
• Solution We have an 4(?2)n, so
• a1 4(?2)1 ?8
• a2 4(?2)2 16
• a3 4(?2)3 ?32
• a4 4(?2)4 64
• a9 4(?2)9 ?2048
• The power (?2)n causes the sign of the terms to
  alternate between positive and negative,
  depending on whether the n is even or odd. This
  kind of sequence is called an alternating
  sequence.

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Sigma Notation

• The Greek letter ? (sigma) can be used to
  simplify notation when the general term of a
  sequence is a formula. For example, the sum of
  the first three terms of the sequence
• ,, , can be named as follows,
  
• using sigma notation, or summation notation
  

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Sigma Notation

• This is read the sum as k goes from 1 to 3 of
  
• The letter k is called the index of summation.
  The index of summation might be a number other
  than 1, and a letter other than k can be used.

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Example

• Find the sum.
• Solution k 2 k 3 k 4
• (?1)232 (?1)333 (?1)434
• 9 (?27) 81
• 63

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• Find the sum.

k 4 k 5 k 6 k
7 k 8
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10.2 Arithmetic Sequences and Series
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Sequences and Summations

• A sequence is an ordered list, possibly infinite,
  of elements.
• We will use the following notation a1, a2, a3, .
  . .
• We also refer to the elements of the sequence as
  terms, and if ak is a term, then k is its index
  or subscript.

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Arithmetic Sequences and Series

• Arithmetic Sequence sequence whose consecutive
  terms have a common difference.
• Example 3, 5, 7, 9, 11, 13, ...
• The terms have a common difference of 2.
• The common difference is the number d.
• Example Is the sequence arithmetic?
  45, 30, 15, 0, 15, 30
• Yes, the common difference is 15
• How do you find any term in this sequence?
• To find any term in an arithmetic sequence, use
  the formula an a1 (n 1)d where d is
  the common difference.

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Example Find a formula for the nth term of the
arithmetic sequence in which the common
difference is 5 and the first term is 3.

• an a1 (n 1)d
• a1 3 d 5
• an 3 (n 1)5

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Example If the common difference is 4 and the
fifth term is 15, what is the 10th term of an
arithmetic sequence?

• an a1 (n 1)d
• We need to determine what the first term is...
• d 4 and a5 15
• a5 a1 (5 1)4 15
• a1 1
• a10 1 (10 1)4
• a10 35

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To find the sum of an arithmetic series
Sn
or
Sn n/2 2a1 (n-1)d
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Example Find the sum of the first 100 terms of
the arithmetic sequence 1, 2, 3, 4, 5, 6, ...

• n 100

5050
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Arithmetic means

• The arithmetic mean of two numbers a and b is
  defined as (ab) / 2. This is the average of a
  b.
• Insert 3 arithmetic means between 2 and 9.
• a1 2 a2 ? a3 ? a4 ? a5 9
• an a1 (n-1) d

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• An explicit formula for a sequence shows how to
  calculate the value of each term from its
  subscript.
• We can also specify a sequence by stating its
  starting value and a recursive formula that tells
  us how to calculate ak from one or more preceding
  values.

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• k 1 2 3 4 5 6 7 8
• ak 7, 11, 15, 19, 23, 27, 31, 35 ...
• Explicit formula ak 7 (k 1) 4 4k 3
• Recursive formula ak ak-1 4

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10.3 Geometric Sequences

• Plus a review of arithmetic sequences

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Definitions

• A sequence is a set of numbers, called terms,
  arranged in some particular order.
• An arithmetic sequence is a sequence with the
  difference between two consecutive terms
  constant.  The difference is called the common
  difference.
• A geometric sequence is a sequence with a common
  ratio, r.

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• Examples Find the common ratio of the following
• 1) 1, 2, 4, 8, 16, ...
• r 2
• 2) 27, 9, 3, 1, 1/3, ...
• r 1/3
• 3) 3, 6, 12, 24, 48, ...
• r 2
• 4) 1/2, -1, 2, -4, 8, ...
• r -2

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Examples Find the next term in each of the
previous sequences.

• 1) 1, 2, 4, 8, 16, ...
• 32
• 2) 27, 9, 3, 1, 1/3, ...
• 1/9
• 3) 3, 6, 12, 24, 48, ...
• 96
• 4) 1/2, -1, 2, -4, 8, ...
• -16

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• Let's play guess the sequence! I give you a
• sequence and you guess the type. 
• 3, 8, 13, 18, 23, . . .  
• 1, 2, 4, 8, 16, . . .
• 24, 12, 6, 3, 3/2, 3/4, . . . 
• 55, 51, 47, 43, 39, 35, . . .
• 2, 5, 10, 17, . . .  
• 1, 4, 9, 16, 25, 36, . . .

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Answers!

• 1) Arithmetic, the common difference d 5
• 2) Geometric, the common ratio r 2
• 3) Geometric, r 1/2
• 4) Arithmetic, d -4
• 5) Neither, why? (How about no common difference
  or ratio!)
• 6) Neither again! (This looks familiar, could it
  be from geometry?)

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This is important!

• Arithmetic formula      
• an    a1    (n - 1)d
• an  is the nth term, a1 is the first term, and d
  is the common difference.
• Geometric formula          
• an  a1 . r (n - 1)
• an is the nth term, a1 is the first term, and r
  is the common ratio.

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Sample problems

• Find the first four terms and state whether the
  sequence is arithmetic, geometric, or neither.
• 1)  an 3n 2   
• 2) an n2 1  
• 3)  an 32n                              

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Answers

• 1)  an 3n 2  
• To find the first four terms, in a row, replace
  n with 1, then 2, then 3 and 4
• Answer 5, 8, 11, 14         
• The sequence is arithmetic!  d 3

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• 2) an n2 1
• To find the first four terms, do the same as
  above!
• Answer  2, 5, 10, 17         
• The sequence is neither.  Why?

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• 3)  an 32n 
• Answer  6, 12, 24, 48        
• The sequence is geometric with
• r 2

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• Find a formula for each sequence.
• 1)  2, 5, 8, 11, 14, . . .
• It is arithmetic! 
• an 2 (n - 1)3 and simplifying yields   an
  3n -1

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• Find the indicated term of the sequence.
• 1)  sequence is arithmetic with t1 5 and t7
  29.  
• Find t53
•    29 5 6d
•                                          
• 24 6d   means d 4 t53 5 52.4 213

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    How to find the sum of a finite Geometric
Series 

• Sn   a1(1 - rn)/(1 - r)
• To find the sum of a finite geometric series,
  you need to know three things  the first term,
  how many terms to add and the common ratio!!

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• Definition
• geometric series - the expression formed by
  adding the terms of a geometric sequence.
• Finding the Sum of the First n Terms of a
  Geometric Sequence.
• Use Sn a1(1 - rn)/(1 - r), Sn is the sum of
  the first n terms.
• Substitute the n, a, and r values into Sn a1(1
  - rn)/(1 - r).
• Simplify to find the sum.

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• Example
• Find the sum of the first 10 terms of the
• geometric series  9 36 144 576 ...
• Answer Sn a1(1 - rn)/(1 - r)
• Sn 9(1- 410)/(1-4)
• Sn 9(-1048575)/(-3)
• Sn 3,145,725

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• Example
• Find the sum of the first 10 terms of the
• geometric series   4 12 36 108 ...
• Answer Sn a1(1 - rn)/(1 - r)
• Sn 4(1- 310)/(1-3)
• Sn 4(-59,048)/(-2)
• Sn 118,096

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Geometric Mean

• The geometric mean between two numbers is the
  square root of the product of the two numbers.
• The geometric mean between 20 45 v900 30

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Finding the sum of an infinite geometric series
If r lt 1, then the infinite geometric series
has the sum
Find the sum of the terms in the infinite
geometric series 1/2 1/4 1/8 1/16
So,
1
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10.5 The Binomial Theorem
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The Binomial Theorem (Binomial expansion)

• (a b)1 1a 1b

coefficient
(a b)2 (a b)(a b) 1a2 2ab
1b2
(a b)3 (a b)(a b)(a b)
1a3 3a2b 3ab2 1b3
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(a b)4 (a b)(a b)(a b)(a b)
1a4 4a3b 6a2b2 4ab31b4
The Binomial Theorem (Binomial expansion)

• Take out the coefficients of each expansion.

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Can you guess the expansion of (a b)5 without
timing out the factors ?
The Binomial Theorem (Binomial expansion)

• (a b)5 1a5 5a4b 10a3b2 10a2b35ab41b4

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The Binomial Theorem (Binomial expansion)
Points to be noticed

• Coefficients are arranged in a Pascal triangle.
• Summation of the indices of each term is equal to
  the power (order) of the expansion.
• The first term of the expansion is arranged in
  descending order after the expansion.
• The second term of the expansion is arranged in
  ascending order order after the expansion.
• Number of terms in the expansion is equal to the
  power of the expansion plus one.

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The Notation of Factorial and Combination

• Factorial
• ---- the product of the first n positive
  integers
• i.e. n! n(n-1)(n-2)(n-3).
• 0!is defined to be 1.
• i.e. 0! 1

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10.6 Permutations
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Fundamental counting principlem1m2m3mk

• At a restaurant, one can choose from 6 salads, 3
  meats, 4 vegetables, and 2 desserts. How many
  different salad-meat-vegetable-dessert
  combinations?
• 6342 144

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Permutations

• A permutation is an ordered arrangement of the
  elements of some set S
• Let S a, b, c
• c, b, a is a permutation of S
• b, c, a is a different permutation of S
• An r-permutation is an ordered arrangement of r
  elements of the set
• A?, 5?, 7?, 10?, K? is a 5-permutation of the set
  of cards
• The notation for the number of r-permutations
  P(n,r)
• The poker hand is one of P(52,5) permutations

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Permutations

• Number of poker hands (5 cards)
• P(52,5) 5251504948 311,875,200
• Number of (initial) blackjack hands (2 cards)
• P(52,2) 5251 2,652
• r-permutation notation P(n,r)
• The poker hand is one of P(52,5) permutations

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r-permutations example

• How many ways are there for 5 people in this
  class to give presentations?
• There are 27 students in the class
• P(27,5) 2726252423 9,687,600
• Note that the order they go in does matter in
  this example!

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Permutation formula proof

• There are n ways to choose the first element
• n-1 ways to choose the second
• n-2 ways to choose the third
• n-r1 ways to choose the rth element
• By the product rule, that gives us
• P(n,r) n(n-1)(n-2)(n-r1)

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Permutations vs. r-permutations

• r-permutations Choosing an ordered 5 card hand
  is P(52,5)
• When people say permutations, they almost
  always mean r-permutations
• But the name can refer to both
• Permutations Choosing an order for all 52 cards
  is P(52,52) 52!
• Thus, P(n,n) n!

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• How many permutations of a, b, c, d, e, f, g
  end with a?
• Note that the set has 7 elements
• The last character must be a
• The rest can be in any order
• Thus, we want a 6-permutation on the set b, c,
  d, e, f, g
• P(6,6) 6! 720

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• How many ways are there to sit 6 people around a
  circular table, where seatings are considered to
  be the same if they can be obtained from each
  other by rotating the table?
• First, place the first person in the north-most
  chair
• Only one possibility
• Then place the other 5 people
• There are P(5,5) 5! 120 ways to do that
• By the product rule, we get 1120 120
• Alternative means to answer this
• There are P(6,6)720 ways to seat the 6 people
  around the table
• For each seating, there are 6 rotations of the
  seating
• Thus, the final answer is 720/6 120

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• A baseball team consists of nine players. Find
  the number of ways of arranging the first four
  positions in the batting order if the pitcher is
  excluded.
• P(8,4) 1680

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• If 8 basketball teams are in a tournament, find
  the number of different ways that first, second,
  third place can be decided, assuming ties are
  not allowed.
• P(8,3) 336

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10.7 Distinguished Permutations Combinations
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Permutations vs. Combinations

• Both are ways to count the possibilities
• The difference between them is whether order
  matters or not
• Consider a poker hand
• A?, 5?, 7?, 10?, K?
• Is that the same hand as
• K?, 10?, 7?, 5?, A?
• Does the order the cards are handed out matter?
• If yes, then we are dealing with permutations
• If no, then we are dealing with combinations

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Combinations

• What if order doesnt matter?
• In poker, the following two hands are equivalent
• A?, 5?, 7?, 10?, K?
• K?, 10?, 7?, 5?, A?
• The number of r-combinations of a set with n
  elements, where n is non-negative and 0rn is

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Distinguishable Permutations

• Of two arrangements of objects, one arrangement
  cannot be obtained from the other by rearranging
  like objects.
• Thus, BBBWR BRBWB are distinguishable
  permutations.
• If, in a collection of n objects, n are alike of
  one kind, n2 are alike of another kind, ,nk are
  alike of a further kind, the of distinguishable
  permutations is

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• Find the number of distinguishable permutations
  of the letters in the word MISSISSIPPI.

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Combinations example

• How many different poker hands are there (5
  cards)?
• How many different (initial) blackjack hands are
  there?

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Example

• A little league baseball squad has six
  outfielders, seven infielders, five pitchers, and
  two catchers. Each outfielder can play any of
  the three outfield positions, and each infielder
  can play any of the four infield positions. In
  how many ways can a team of nine players be
  chosen?

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A last note on combinations

• An alternative (and more common) way to denote an
  r-combination
• Ill use C(n,r) whenever possible, as it is
  easier to write in PowerPoint

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10.8 Probability

• Objective
• Students will be introduced to both theoretical
  and experimental probabilities. Students will
  also find probabilities of mutually exclusive
  events.

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The probability of an event is a number between
0 and 1 that indicates the likelihood the event
will occur.
There are two types of probability theoretical
and experimental.
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The theoretical probability of an event is often
simply called the probability of the event.
When all outcomes are equally likely,
the theoretical probability that an event A
will occur is
number of outcomes in A
P (A)
outcomes in event A
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You roll a six-sided die whose sides are numbered
from 1 through 6.
Find the probability of rolling a 4.
SOLUTION
Only one outcome corresponds to rolling a 4.
number of ways to roll a 4
P (rolling a 4)
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You roll a six-sided die whose sides are numbered
from 1 through 6.
Find the probability of rolling an odd number.
SOLUTION
Three outcomes correspond to rolling an odd
number rolling a 1, 3, or a 5.
number of ways to roll an odd number
P (rolling odd number)
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You roll a six-sided die whose sides are numbered
from 1 through 6.
Find the probability of rolling a number less
than 7.
SOLUTION
All six outcomes correspond to rolling a number
less than 7.
number of ways to roll less than 7
P (rolling less than 7 )
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You put a CD that has 8 songs in your CD player.
You set the player to play the songs at random.
The player plays all 8 songs without repeating
any song.
What is the probability that the songs are
played in the same order they are listed on the
CD?
Help
SOLUTION
There are 8! different permutations of the 8
songs. Of these, only 1 is the order in which
the songs are listed on the CD. So, the
probability is
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You put a CD that has 8 songs in your CD player.
You set the player to play the songs at random.
The player plays all 8 songs without repeating
any song.
You have 4 favorite songs on the CD. What is the
probability that 2 of your favorite songs are
played first, in any order?
Help
SOLUTION
There are 8C2 different combinations of 2 songs.
Of these, 4C2 contain 2 of your favorite songs.
So, the probability is
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Sometimes it is not possible or convenient to
find thetheoretical probability of an event. In
such cases youmay be able to calculate an
experimental probabilityby performing an
experiment, conducting a survey, orlooking at
the history of the event.
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In 1998 a survey asked Internet users for their
ages. The results are shown in the bar graph.
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Find the experimental probability that a randomly
selected Internet user is at most 20 years old.
SOLUTION
The number of people surveyed was 1636 6617
3693 491 6 12,443.
Of the people surveyed, 16 36 are at most 20
years old.
So, the probability is
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Find the experimental probability that a randomly
selected Internet user is at least 41 years old.
Given that 12,443 people were surveyed.
SOLUTION
Of the people surveyed, 3693 491 6
4190 are at least 41 years old.
So, the probability is
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You throw a dart at the board shown. Your dart
is equallylikely to hit any point inside
the square board. Are you more likely to get 10
points or 0 points?
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Are you more likely to get 10 points or 0 points?
SOLUTION
area of smallest circle
P (10 points)
area outside largest circle
P (0 points)
You are more likely to get 0 points.
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10.8 (day 2) Probability of Compound Events

• P. 724

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Mutually Exclusive Events
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Intersection of A B
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• To find P(A or B) you must consider what
  outcomes, if any, are in the intersection of A
  and B.
• If there are none, then A and B are mutually
  exclusive events and P(A or B) P(A)P(B)
• If A and B are not mutually exclusive, then the
  outcomes in the intersection or A B are counted
  twice when P(A) P(B) are added.
• So P(A B) must be subtracted once from the sum

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EXAMPLE 1

• One six-sided die is rolled.
• What is the probability of rolling a multiple of
  3 or 5?
• P(A or B) P(A) P(B) 2/6 1/6 1/2
• 0.5

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EXAMPLE 2

• One six-sided die is rolled. What is the
  probability of rolling a multiple of 3 or a
  multiple of 2?
• A Mult 3 2 outcomes
• B mult 2 3 outcomes
• P(A or B) P(A) P(B) P(AB)
• P(A or B) 2/6 3/6 1/6
• 2/3 0.67

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EXAMPLE 3

• In a poll of high school juniors, 6 out of 15
  took a French class and 11 out of 15 took a math
  class.
• Fourteen out of 15 students took French or math.
• What is the probability that a student took both
  French and math?

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• A French
• B Math
• P(A) 6/15, P(B) 11/15, P(AorB) 14/15
• P(A or B) P(A) P(B) P(AB)
• 14/15 6/15 11/15 P(A B)
• P(A B) 6/15 11/15 14/15
• P(A B) 3/15 1/5 .20

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Using complements to find Probability

• The event A, called the complement of event A,
  consists of all outcomes that are not in A.
• The notation A is read A prime.

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Probability of the complement of an event

• The probability of the complement of A is
• P(A) 1 - P(A)

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EXAMPLE 4

• A card is randomly selected from a standard deck
  of 52 cards.
• Find the probability of the given event.
• a. The card is not a king.
• P(K) 4/52 so P(K)
• 48/52 0.923

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• b. The card is not an ace or a jack.
• P(not ace or Jack) 1-(P(4/52 4/52))
• 1- 8/52
• 44/52 0.846

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Probability of Independent and Dependent Events
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• Two events are Independent if the occurrence of 1
  has no effect on the occurrence of the other. (a
  coin toss 2 times, the first toss has no effect
  on the 2nd toss)

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Probability of Two Independent Events(can be
extended to probability of 3 or more ind. events)

• A B are independent events then the probability
  that both A B occur is
• P(A and B) P(A) P(B)

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BASEBALL

• During the 1997 baseball season, the Florida
  Marlins won 5 out of 7 home games and 3 out of 7
  away games against the San Francisco Giants.
  During the 1997 National League Division Series
  with the Giants, the Marlins played the first two
  games at home and the third game away. The
  Marlins won all three games.
• Estimate the probability of this happening. _
  Source The Florida Marlins

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• Let A, B, C be winning the 1st, 2nd, 3rd
  games
• The three events are independent and have
  experimental probabilities based on the regular
  season games.
• P(ABC) P(A)P(B)P(C)
• 5/7 5/7 3/7 75/343
• .219

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PROBABILITIES OF DEPENDENT EVENTS

• Two events A and B are dependent events if the
  occurrence of one affects the occurrence of the
  other.
• The probability that B will occur given that A
  has occurred is called the conditional
  probability of B given A and is written P(BA).

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Probability of Dependent Events

• If A B are dependant events, then the
  probability that both A B occur is
• P(AB) P(A) P(B/A)

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Comparing Dependent and Independent Events

• You randomly select two cards from a standard
  52-card deck. What is the probability that the
  first card is not a face card (a king, queen, or
  jack) and the second card is a face card if
• (a) you replace the first card before selecting
  the second, and
• (b) you do not replace the first card?

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• (A) If you replace the first card before
  selecting the second card, then A and B are
  independent events. So, the probability is
• P(A and B) P(A) P(B) 40 12 30
  52 52 169
• 0.178
• (B) If you do not replace the first card before
  selecting the second card, then A and B are
  dependent events. So, the probability is
• P(A and B) P(A) P(BA) 4012 40
  52 51 221
• 0.181

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• http//www.youtube.com/watch?vXllKz2cX73Q