Chapter 5 Probability

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Chapter 5 Probability

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Title: Chapter 5 Probability

1
Chapter 5Probability

5.3
The Multiplication Rule

2
Section 5.1 gave rules for calculating
probabilities for the union of two events E and
F
3
Section 5.1 gave rules for calculating
probabilities for the union of two events E and
F P(E or F) P(E) P(F) P(E and F)
4
Section 5.1 gave rules for calculating
probabilities for the union of two events E and
F P(E or F) P(E) P(F) P(E and
F) Section 5.2 gives rules for calculating the
intersection of two events E and F
5
Section 5.1 gave rules for calculating
probabilities for the union of two events E and
F P(E or F) P(E) P(F) P(E and
F) Section 5.2 gives rules for calculating the
intersection of two events E and F
(E and F)
6

EXAMPLE Roll two dice and add the rolls.

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?

8

EXAMPLE Get a total of six
9

EXAMPLE Get a total of six
10

EXAMPLE Get a total of six
11

EXAMPLE Get a total of six
12

EXAMPLE Get a total of six
13

EXAMPLE Get a total of six
14

EXAMPLE Get a total of six P(Get a total
of six) 5/36
15

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?
P(get a total of six) 5/36

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?
P(get a total of six) 5/36
What is the probability of getting a total of six
if we know that the first die is a four?

17

EXAMPLE Get a total of six given first die is a
four
18

EXAMPLE Get a total of six given first die is a
four
19

EXAMPLE Get a total of six given first die is a
four
20

EXAMPLE Get a total of six given first die is a
four
21

EXAMPLE Get a total of six given first die is a
four
22

EXAMPLE Get a total of six given first die is a
four
23

EXAMPLE Get a total of six given first die is a
four
24

EXAMPLE Get a total of six given first die is a
four There are six simple events such that
the first roll is equal to 4.
25

EXAMPLE Get a total of six given first die is a
four There are six simple events such that
the first roll is equal to 4. Each of these is
equally likely to happen.
26

EXAMPLE Get a total of six given first die is a
four There are six simple events such that
the first roll is equal to 4. Each of these is
equally likely to happen. Only one of these
events results in a sum of six.
27

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?
P(get a total of six) 5/36
What is the probability of getting a total of six
if we know that the first die is a four?
P(get a total of six first die is a four)
1/6

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?
P(get a total of six) 5/36
What is the probability of getting a total of six
if we know that the first die is a four?
P(get a total of six first die is a four)
1/6
What is the probability of getting a total of
six if we know that the first die is a six?

29

EXAMPLE Get a total of six given first die is a
six
30

EXAMPLE Get a total of six given first die is a
six
31

EXAMPLE Get a total of six given first die is a
six
32

EXAMPLE Get a total of six given first die is a
six
33

EXAMPLE Get a total of six given first die is a
six
34

EXAMPLE Get a total of six given first die is a
six
35

EXAMPLE Get a total of six given first die is a
six
36

EXAMPLE Get a total of six given first die is a
six There are six simple events where the
first die roll is a six.
37

EXAMPLE Get a total of six given first die is a
six There are six simple events where the
first die roll is a six. None of these events has
a sum equal to six.
38

EXAMPLE Roll two dice and add the rolls.
What is the probability of getting a total of
six?
P(get a total of six) 5/36
What is the probability of getting a total of six
if we know that the first die is a four?
P(get a total of six first die is a four)
1/6
What is the probability of getting a total of
six if we know that the first die is a six?
P(get a total of six first die is a six) 0

39
These are examples of conditional
probability.
40
These are examples of conditional
probability. Essentially, the idea is that the
probability of an event depends on our knowledge
of related events.
41
These are examples of conditional
probability. Essentially, the idea is that the
probability of an event depends on our knowledge
of related events. If we know that some event E
has occurred, this might change the probability
of F occurring.
42
Conditional Probability
43
Conditional Probability P(F E) is the
probability of event F given event E.
44
Conditional Probability P(F E) is the
probability of event F given event E. It is
the probability of an event F given that event E
has occurred.
45
EXAMPLES
46
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6
47
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6 P(Sum of two dice equal
six First die equals six) 0
48
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6 P(Sum of two dice equal
six First die equals six) 0 P(Sum of two
dice equal six Sum of two dice equal six) ?
49
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6 P(Sum of two dice equal
six First die equals six) 0 P(Sum of two
dice equal six Sum of two dice equal six) 1
50
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6 P(Sum of two dice equal
six First die equals six) 0 P(Sum of two
dice equal six Sum of two dice equal six)
1 P(Sum of two dice equal six Sum of two dice
not equal six) ?
51
EXAMPLES P(Sum of two dice equal six First
die equals four) 1/6 P(Sum of two dice equal
six First die equals six) 0 P(Sum of two
dice equal six Sum of two dice equal six)
1 P(Sum of two dice equal six Sum of two dice
not equal six) 0
52
Multiplication Rule
53
Multiplication Rule The Probability of the event
(E and F) is given by
54
Multiplication Rule The Probability of the event
(E and F) is given by P(F and E) P(F E)
P(E)
55
Multiplication Rule The Probability of the event
(E and F) is given by P(F and E) P(F E)
P(E) also P(F and E) P(E F) P(F)
56
EXAMPLE Using the Multiplication Rule
57
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations
58
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations ? A K Q J 10 9 8 7 6 5 4 3 2

59
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2
60
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8
7 6 5 4 3 2
61
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8
7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
62
EXAMPLE Using the Multiplication Rule A deck
of playing cards consists of 4 suits and 13
denominations ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8
7 6 5 4 3 2 52 cards total ? A K Q J 10 9 8 7
6 5 4 3 2
63
EXAMPLE Using the Multiplication Rule Assume
the deck is randomly shuffled so that each card
is equally likely to be drawn.
64
EXAMPLE Using the Multiplication Rule Assume
the deck is randomly shuffled so that each card
is equally likely to be drawn. Drawing cards with
replacement means that each time we draw a card
it is replaced in the deck and the deck is
reshuffled.
65
EXAMPLE Using the Multiplication Rule Assume
the deck is randomly shuffled so that each card
is equally likely to be drawn. Drawing cards with
replacement means that each time we draw a card
it is replaced in the deck and the deck is
reshuffled. Drawing cards without replacement
means each time we draw a card we keep it out of
the deck when drawing more cards (e.g., poker)
66
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card.
67
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
68
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
69
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
70
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
71
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
72
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 P(E) ? ? A K
Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5
4 3 2
73
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 P(E) 4/52
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2
74
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 P(E) 4/52
1/13 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J
10 9 8 7 6 5 4 3 2
75
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 P(E) 4/52
1/13 0.08 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A
K Q J 10 9 8 7 6 5 4 3 2
76
EXAMPLE Using the Multiplication Rule F Draw
Ace on Second Card (with replacement).
77
EXAMPLE Using the Multiplication Rule F Draw
Ace on Second Card (with replacement). When
drawing with replacement, we are basically
starting over from scratch.
78
EXAMPLE Using the Multiplication Rule F Draw
Ace on Second Card (with replacement). When
drawing with replacement, we are basically
starting over from scratch. P(F) 1/13
79
EXAMPLE Using the Multiplication Rule
80
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the First Card
81
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the First Card F Draw Ace on Second
Card (without replacement).
82

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) ?

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) 3/51

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) 3/51
P(E and F) ?

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) 3/51
P(E and F) (4/52) x (3/51)

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) 3/51
P(E and F) (4/52) x (3/51) 12/2652

EXAMPLE Using the Multiplication Rule
E Draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 4/52
P(F E) 3/51
P(E and F) (4/52) x (3/51) 12/2652
0.005

90
EXAMPLE Using the Multiplication Rule
91
EXAMPLE Using the Multiplication Rule E Do
not draw an Ace on the First Card
92
EXAMPLE Using the Multiplication Rule E Do
not draw an Ace on the First Card F Draw Ace on
Second Card (without replacement).
93

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) ?

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E)

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E) 4/51

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E) 4/51
P(E and F)

100

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E) 4/51
P(E and F) (48/52) x (4/51)

101

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E) 4/51
P(E and F) (48/52) x (4/51) 192/2652

102

EXAMPLE Using the Multiplication Rule
E Do not draw an Ace on the First Card
F Draw Ace on Second Card (without
replacement).
P(E and F) P(F E) P(E)
P(E) 1 4/52 48/52
P(F E) 4/51
P(E and F) (48/52) x (4/51) 192/2652
0.073

103
EXAMPLE Using the Multiplication Rule F Draw
Ace on Second Card (without replacement).
104

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
E First card is an Ace

105

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
E First card is an Ace
( E and F)

106

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
E First card is an Ace
( E and F) First and second cards are Aces

107

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
E First card is an Ace
( E and F) First and second cards are Aces
( E and F)

108

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
E First card is an Ace
( E and F) First and second cards are Aces
( E and F) First card not an Ace and second
card an Ace

109

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)

110

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)

111

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.

112
The event (E and F) is disjoint from the event (E
and F)
113
The event (E and F) is disjoint from the event (E
and F)
F
114
The event (E and F) is disjoint from the event (E
and F)
F
115
The event (E and F) is disjoint from the event (E
and F)
E and F
F
116
The event (E and F) is disjoint from the event (E
and F)
E and F
E and F
F
117
The event (E and F) is disjoint from the event (E
and F) F ( E and F) or ( E and F)
E and F
E and F
F
118
The event (E and F) is disjoint from the event (E
and F) F ( E and F) or ( E and
F) P(F) P( E and F) P( E and F)
E and F
E and F
F
119
The event (E and F) is disjoint from the event (E
and F) F ( E and F) or ( E and
F) P(F) P( E and F) P( E and F) -
addition rule for disjoint events
E and F
E and F
F
120

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.
P(F) P( E and F) P( E and F)

121

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.
P(F) P( E and F) P( E and F)
12/2652

122

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.
P(F) P( E and F) P( E and F)
12/2652 192/2652

123

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.
P(F) P( E and F) P( E and F)
12/2652 192/2652
204 / 2652

124

EXAMPLE Using the Multiplication Rule
F Draw Ace on Second Card (without
replacement).
F ( E and F) or ( E and F)
( E and F) is disjoint from the event ( E and
F)
We cant simultaneously draw an Ace on the
first card and NOT draw an Ace on the first
card.
P(F) P( E and F) P( E and F)
12/2652 192/2652
204 / 2652 1/13

125
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the Second Card (with replacement)
126
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the Second Card (with replacement)
P(E) 1/13
127
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the Second Card (with replacement)
P(E) 1/13 F Draw Ace on Second Card (without
replacement)
128
EXAMPLE Using the Multiplication Rule E Draw
an Ace on the Second Card (with replacement)
P(E) 1/13 F Draw Ace on Second Card (without
replacement) P(F) 1/13
129
EXAMPLE Using the Multiplication
Rule Question Why does this make sense?
130
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned anything.
131
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned
anything. The next card dealt is just as likely
to be an Ace as if we hadnt dealt the first card.
132
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned
anything. The next card dealt is just as likely
to be an Ace as if we hadnt dealt the first
card. If we DO look at the first card, we learn
something and the probability changes.
133
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned
anything. The next card dealt is just as likely
to be an Ace as if we hadnt dealt the first
card. If we DO look at the first card, we learn
something and the probability changes. P(F E)
3/51
134
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned
anything. The next card dealt is just as likely
to be an Ace as if we hadnt dealt the first
card. If we DO look at the first card, we learn
something and the probability changes. P(F E)
3/51 P(F E) 4/51
135
EXAMPLE Using the Multiplication
Rule Question Why does this make
sense? Answer Because if we draw a card and
leave it face down, we havent learned
anything. The next card dealt is just as likely
to be an Ace as if we hadnt dealt the first
card. If we DO look at the first card, we learn
something and the probability changes. P(F E)
3/51 P(F E) 4/51 Were more likely to draw
an Ace on the second card if we know we havent
drawn an Ace on the first card.
136
EXAMPLE Using the Multiplication
Rule Conditional Probability has the
interpretation of updating the probability of an
event happening after you have learned that some
other event has happened.
137
EXAMPLE Using the Multiplication
Rule Conditional Probability has the
interpretation of updating the probability of an
event happening after you have learned that some
other event has happened. E Cubs win the World
Series this year.
138
EXAMPLE Using the Multiplication
Rule Conditional Probability has the
interpretation of updating the probability of an
event happening after you have learned that some
other event has happened. E Cubs win the World
Series this year. F Cubs win the NLCS this year
139
EXAMPLE Using the Multiplication
Rule Conditional Probability has the
interpretation of updating the probability of an
event happening after you have learned that some
other event has happened. E Cubs win the World
Series this year. F Cubs win the NLCS this
year P(E F) 0.
140
Sometimes the information from the event E tells
us nothing at all about the event F.
141
Sometimes the information from the event E tells
us nothing at all about the event F. F
Yankees win the World Series this year.
142
Sometimes the information from the event E tells
us nothing at all about the event F. F
Yankees win the World Series this year. E
Germany wins the Womens World Cup of Soccer
this year.
143
Sometimes the information from the event E tells
us nothing at all about the event F. F
Yankees win the World Series this year. E
Germany wins the Womens World Cup of Soccer
this year. Knowing E tells us nothing at all
about F.
144
Sometimes the information from the event E tells
us nothing at all about the event F. F
Yankees win the World Series this year. E
Germany wins the Womens World Cup of Soccer
this year. Knowing E tells us nothing at all
about F. P(F E) ?
145
Sometimes the information from the event E tells
us nothing at all about the event F. F
Yankees win the World Series this year. E
Germany wins the Womens World Cup of Soccer
this year. Knowing E tells us nothing at all
about F. P(F E) P(F)
146
INDEPENDENT EVENTS
147
INDEPENDENT EVENTS Two events E and F are
independent if the occurrence of event E does not
affect the probability of event F.
148
INDEPENDENT EVENTS Two events E and F are
independent if the occurrence of event E does not
affect the probability of event F. P(F E)
P(F)
149
NOTE
150
NOTE P(F E) P(F) If and only if P(E
F) P(E)
151
NOTE P(F E) P(F) If and only if P(E
F) P(E) Thus to show independence of two
events, we can show one or the other.
152
DEPENDENT EVENTS
153
DEPENDENT EVENTS Two events are dependent if
the occurrence of event E in affects the
probability of event F.
154
DEPENDENT EVENTS Two events are dependent if
the occurrence of event E in affects the
probability of event F. P(F E) is not
equal to P(F)
155
EXAMPLE Independent and Dependent Events
156
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card
157
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement)
158
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement) Are E and F
dependent or independent?
159
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement) Are E and F
dependent or independent? P(F) 1/13
160
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement) Are E and F
dependent or independent? P(F) 1/13 P(F E) ?
161
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement) Are E and F
dependent or independent? P(F) 1/13 P(F E)
1/13
162
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (with replacement) Are E and F
dependent or independent? P(F) 1/13 P(F E)
1/13 E and F are independent events
163
Experiments which are done by sampling with
replacement result in independent outcomes.
164
EXAMPLE Independent and Dependent Events
165
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card
166
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (without replacement)
167
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (without replacement) Are E and
F dependent or independent?
168
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (without replacement) Are E and
F dependent or independent? P(F) 4/52
169
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (without replacement) Are E and
F dependent or independent? P(F) 4/52 P(F E)
3/51
170
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw an Ace on
the second card (without replacement) Are E and
F dependent or independent? P(F) 4/52 P(F E)
3/51 E and F are dependent events
171
Experiments which are done by sampling without
replacement result in dependent outcomes.
172
Experiments which are done by sampling without
replacement result in dependent outcomes. By
sampling and not replacing a member of the
population were changing the composition of the
remaining population.
173
Experiments which are done by sampling without
replacement result in dependent outcomes. By
sampling and not replacing a member of the
population were changing the composition of the
remaining population. However, if the population
is large in comparison to the sample (sample is practically independent.
174
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card.
175
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
176
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
177
EXAMPLE Using the Multiplication Rule E Draw
Ace on First Card. ? A K Q J 10 9 8 7 6 5 4 3
2 ? A K Q J 10 9 8 7 6 5 4 3 2 P(E) 4/52 ? A
K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6
5 4 3 2
178
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw a Club on
the first card ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8
7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
179
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw a Club on
the first card ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9
8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
180
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card F Draw a Club on
the first card ? A K Q J 10 9 8 7 6 5 4 3 2
? A K Q J 10 9 8 7 6 5 4 3 2 P(F) 13/52
1/4 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10
9 8 7 6 5 4 3 2
181
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card
182
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13
183
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card
184
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4
185
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events?
186
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(E F)
?
187
EXAMPLE Independent and Dependent Events ? A
K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5
4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J
10 9 8 7 6 5 4 3 2
188
EXAMPLE Independent and Dependent Events F
draw a club ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K
Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4
3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
189
EXAMPLE Independent and Dependent Events F
draw a club ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K
Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4
3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
190
EXAMPLE Independent and Dependent Events F
draw a club ? A K Q J 10 9 8 7 6 5 4 3 2
191
EXAMPLE Independent and Dependent Events F
draw a club E draw an Ace ? A K Q J 10 9
8 7 6 5 4 3 2
192
EXAMPLE Independent and Dependent Events F
draw a club E draw an Ace ? A K Q J 10 9
8 7 6 5 4 3 2
193
EXAMPLE Independent and Dependent Events F
draw a club E draw an Ace P(E F)
1/13 ? A K Q J 10 9 8 7 6 5 4 3 2
194
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(E F)
1/13
195
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(E F)
1/13 P(E)
196
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(E F)
1/13 P(E) Thus E and F are independent
events.
197
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(F E)
?
198
EXAMPLE Independent and Dependent Events ? A
K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5
4 3 2 ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K Q J
10 9 8 7 6 5 4 3 2
199
EXAMPLE Independent and Dependent Events E
draw an Ace ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K
Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4
3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
200
EXAMPLE Independent and Dependent Events E
draw an Ace ? A K Q J 10 9 8 7 6 5 4 3 2 ? A K
Q J 10 9 8 7 6 5 4 3 2 ? A K Q J 10 9 8 7 6 5 4
3 2 ? A K Q J 10 9 8 7 6 5 4 3 2
201
EXAMPLE Independent and Dependent Events E
draw an Ace ? A ? A ? A ? A
202
EXAMPLE Independent and Dependent Events E
draw an Ace F draw a club ? A ? A ? A ? A

203
EXAMPLE Independent and Dependent Events E
draw an Ace F draw a club ? A ? A ? A ? A

204
EXAMPLE Independent and Dependent Events E
draw an Ace F draw a club ? A ? A ? A ?
A P(F E) 1/4
205
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(F E)
1/4
206
EXAMPLE Independent and Dependent Events E
Draw an Ace on the first card P(E) 1/13 F
Draw a Club on the first card P(F) 1/4 Are E
and F dependent or independent events? P(F E)
1/4 P(F) Thus F and E are independent.
207
P(FE) P(F) if and only if P(EF) P(E)
208
P(FE) P(F) if and only if P(EF) P(E)
Thus we need only demonstrate one or the other
to show independence.
209
MULTIPLICATION RULE FOR INDPENDENT EVENTS
210
MULTIPLICATION RULE FOR INDPENDENT EVENTS If
events E and F are independent
211
MULTIPLICATION RULE FOR INDPENDENT EVENTS If
events E and F are independent P(E and F)
P(F) P(E )
212

Recall that the multiplication rule which is
true for any two events E and F is

213

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)

214

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,

215

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,
P(F E) P(F)

216

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,
P(F E) P(F)
Thus if E is independent of F,

217

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,
P(F E) P(F)
Thus if E is independent of F,
P(F E) P(F)

218

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,
P(F E) P(F)
Thus if E is independent of F,
P(F E) P(E) P(F) P(E)

219

Recall that the multiplication rule which is
true for any two events E and F is
P(E and F) P(F E) P(E)
If E is independent of F,
P(F E) P(F)
Thus if E is independent of F,
P(F E) P(E) P(F) P(E)
P(E and F)

220
MORE THAN TWO INDEPENDENT EVENTS
221
MORE THAN TWO INDEPENDENT EVENTS E1, E2, . . .
, En are independent events if any two of them
are
222
MORE THAN TWO INDEPENDENT EVENTS E1, E2, . . .
, En are independent events if any two of them
are E1 independent of E2
223
MORE THAN TWO INDEPENDENT EVENTS E1, E2, . . .
, En are independent events if any two of them
are E1 independent of E2 E1 independent of E3
224
MORE THAN TWO INDEPENDENT EVENTS E1, E2, . . .
, En are independent events if any two of them
are E1 independent of E2 E1 independent of E3 E2
independent of E3
225
MORE THAN TWO INDEPENDENT EVENTS E1, E2, . . .
, En are independent events if any two of them
are E1 independent of E2 E1 independent of E3 E2
independent of E3 etc.
226
MULTIPLICATION RULE FOR n INDPENDENT EVENTS
227
MULTIPLICATION RULE FOR n INDPENDENT EVENTS If
events E1, E2, . . . , En are independent n
independent events
228
MULTIPLICATION RULE FOR n INDPENDENT EVENTS If
events E1, E2, . . . , En are independent n
independent events P(E1 and E2 and . . . and En
) P(E1) P(E2) . . . P(En)
229

EXAMPLE Multiplication for Independent Events
The probability that a randomly selected US
female aged 20 years old will survive the year is
.9995.

230

EXAMPLE Multiplication for Independent Events
The probability that a randomly selected US
female aged 20 years old will survive the year is
.9995.
What is the probability that four randomly
selected 20 year old females will survive the
year?

231
EXAMPLE Multiplication for Independent Events E1
1st female survives
232
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives
233
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives
234
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives
235
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other.
236
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive
237
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive P (E1 and E2 and E3 and E4)
?
238
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive P (E1 and E2 and E3 and E4)
P (E1) P (E2) P (E3) P (E4)
239
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive P (E1 and E2 and E3 and E4)
P (E1) P (E2) P (E3) P (E4) (.9995) (.9995)
(.9995) (.9995)
240
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive P (E1 and E2 and E3 and E4)
P (E1) P (E2) P (E3) P (E4) (.9995) (.9995)
(.9995) (.9995) .99954
241
EXAMPLE Multiplication for Independent Events E1
1st female survives E2 2nd female
survives E3 3rd female survives E4 4th
female survives These events are independent of
each other. (E1 and E2 and E3 and E4) event
all four survive P (E1 and E2 and E3 and E4)
P (E1) P (E2) P (E3) P (E4) (.9995) (.9995)
(.9995) (.9995) .99954 0.9980015
242
RULE Multiplication for Independent
Events Suppose we have n independent events E1,
E2 , . . . , En
243
RULE Multiplication for Independent
Events Suppose we have n independent events E1,
E2 , . . . , En Suppose each of these
events has probability p P(E1) P(E1) . .
. P(En) p
244
RULE Multiplication for Independent
Events Suppose we have n independent events E1,
E2 , . . . , En Suppose each of these
events has probability p P(E1) P(E1) . .
. P(En) p Then P(E1 and E1 and . . . and
En) pn
245

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.

246

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.

247

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?

248

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four

249

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four

250

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four
E3 3rd die a four

251

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four

252

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four
E5 5th die a four

253

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four
E5 5th die a four
E (E1 and E2 and E3 and E4 and E5)

254

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) ?
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four
E5 5th die a four
E (E1 and E2 and E3 and E4 and E5)
P(Ei ) 1/6 for all five events

255

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) (1/6)5
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four
E5 5th die a four
E (E1 and E2 and E3 and E4 and E5)
P(Ei ) 1/6 for all five events

256

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E Get 5 fours.
P(E) (1/6)5 0.0001286
E1 1st die a four E2 2nd die a four
E3 3rd die a four E4 4th die a four
E5 5th die a four
E (E1 and E2 and E3 and E4 and E5)
P(Ei ) 1/6 for all five events

257

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.

258

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E We get at least one four.

259

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E We get at least one four.
In other words, we get either 1 four, or 2 fours,
or 3 fours, or 4 fours, or 5 fours.

260

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E We get at least one four.
In other words, we get either 1 four, or 2 fours,
or 3 fours, or 4 fours, or 5 fours.
What is the probability of E?

261

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E We get at least one four.
In other words, we get either 1 four, or 2 fours,
or 3 fours, or 4 fours, or 5 fours.
What is the probability of E?
P(1 four) P(2 fours) P(3 fours) P(4 fours)
P(5 fours)

262

EXAMPLE Multiplication for Independent Events
Experiment Roll 5 dice.
E We get at least one four.
In other words, we get either 1 four, or 2 fours,
or 3 fours, or 4 fours, or 5 fours.
What is the probability of E?
P(1 four) P(2 fours) P(3 fours) P(4 fours)
P(5 fours)
It is easier to calculate probability of E We
get no fours.

263