Download Guided Practice - Social Circle City Schools

Introduction
In probability, events are either dependent or
independent. Two events are independent if the
occurrence or non-occurrence of one event has no effect
on the probability of the other event. If two events are
independent, then you can simply multiply their
individual probabilities to find the probability that both
events will occur. If events are dependent, then the
outcome of one event affects the outcome of another
event. So it is important to know whether or not two
events are independent.
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7.1.3: Understanding Independent Events
Introduction, continued
Two events A and B are independent if and only if
P(A and B) = P(A) • P(B). But sometimes you only know
two of these three probabilities and you want to find the
third. In such cases, you can’t test for independence, so
you might assess the situation or nature of the
experiment and then make an assumption about
whether or not the events are independent. Then, based
on your assumption, you can find the third probability in
the equation.
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7.1.3: Understanding Independent Events
Key Concepts
• Two events A and B are independent if and only if
they satisfy the following test:
P(A and B) = P(A) • P(B)
• Using set notation, the test is P ( A Ç B ) = P ( A) · P ( B ) .
• Sometimes it is useful or necessary to make an
assumption about whether or not events are
independent, based on the situation or nature of the
experiment. Then any conclusions or answers are
based on the assumption.
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7.1.3: Understanding Independent Events
Key Concepts, continued
• In a uniform probability model, all the outcomes of an
experiment are assumed to be equally likely, and the
probability of an event E, denoted P(E), is given by
number of outcomes in E
P (E ) =
.
number of outcomes in the sample space
• When this definition of probability is used and the
relevant probabilities are known, then the definition of
independent events can be used to determine, verify,
or prove that two events are dependent or
independent.
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7.1.3: Understanding Independent Events
Key Concepts, continued
• The relative frequency of an event is the number of
times it occurs divided by the number of times the
experiment is performed (called trials) or the number
of observations:
number of times event occurs
relative frequency =
number of trials or observations
• Relative frequency can be used to estimate probability
in some cases where a uniform probability model
does not seem appropriate. Such cases include data
collected by surveys or obtained by observations.
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7.1.3: Understanding Independent Events
Key Concepts, continued
• When relative frequency is used to estimate the
relevant probabilities, then the definition of
independent events can be used to determine whether
two events seem to be dependent or independent,
based on the data.
• Probability and relative frequency are related as
follows:
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7.1.3: Understanding Independent Events
Key Concepts, continued
• The probability of an event can be used to predict
its relative frequency if the experiment is performed
a large number of times. For example, the
probability of getting a 3 by rolling a fair die is
1
.
6
So if you roll a fair die 6,000 times, it is reasonable
to predict that you will get a 3 about 1,000 times, or
1 of the number of times you roll the die. If you roll
6
a die 6,000 times, the number of times you get a 3
might not be 1,000.
7.1.3: Understanding Independent Events
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Key Concepts, continued
• Relative frequency can be used to predict the
probability of an event. In general, as the number
of trials or observations increases, the prediction
becomes stronger. For example, suppose you ask
40 people for their favorite ice cream flavor. If 8 say
chocolate, then you can predict that the probability
of a randomly selected person saying chocolate is
8
, or 20%. But 40 people make up a small
40
sample, so this is not a very strong prediction.
7.1.3: Understanding Independent Events
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Key Concepts, continued
Now suppose you ask 4,000 people who are
randomly selected using a good sampling method.
If 740 say chocolate, then you can predict that the
probability of a randomly selected person saying
740
chocolate is
, or 18.5%; this is a stronger
4000
prediction.
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7.1.3: Understanding Independent Events
Key Concepts, continued
• Also remember the Addition Rule: If A and B are
any two events, then the probability of A or B,
denoted P(A or B), is given by
P(A or B) = P(A) + P(B) – P(A and B).
• Using set notation, the rule is
P ( A È B ) = P ( A) + P ( B ) - P ( A Ç B ) .
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7.1.3: Understanding Independent Events
Common Errors/Misconceptions
• thinking that the actual relative frequency of an event
will equal the probability of the event; for example,
thinking 100 tosses of a fair coin will yield 50 heads and
50 tails
• thinking that a probability based on actual relative
frequency is a “true fact;” that is, not understanding that
the probability is just a number that is only as good as
the assumptions and statistical sampling methods it is
based on
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7.1.3: Understanding Independent Events
Common Errors/Misconceptions, continued
• thinking that a probability based on past actual relative
frequency and obtained using reasonable assumptions
and sound statistical sampling methods can be used to
guarantee the actual future relative frequency of an
event; that is, not realizing that even a probability
obtained by the best possible means is only a predictor,
never a guarantee
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7.1.3: Understanding Independent Events
Guided Practice
Example 2
Trevor tosses a coin 3 times. Consider the following events.
A: The first toss is heads.
B: The second toss is heads.
C: There are exactly 2 consecutive heads.
For each of the following pairs of events, determine if the
events are independent.
A and B (This is A Ç B in set notation.)
A and C (This is A Ç C in set notation.)
B and C (This is B Ç C in set notation.)
7.1.3: Understanding Independent Events
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Guided Practice: Example 2, continued
1. List the sample space.
Sample space = {HHH, HHT, HTH, HTT, THH, THT,
TTH, TTT}
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
2. Use the sample space to determine the
relevant probabilities.
P ( A) =
P (B) =
P (C ) =
4
8
4
8
2
8
=
=
=
1
2
1
2
1
4
There are 4 outcomes with
heads first.
There are 4 outcomes with
heads second.
There are 2 outcomes with
exactly 2 consecutive heads.
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
P ( A Ç B) =
P (A Ç C) =
P (B Ç C ) =
2
8
=
1
4
1
8
2
8
=
1
4
There are 2 outcomes with
heads first and heads
second.
There is 1 outcome with
heads first and exactly 2
consecutive heads.
There are 2 outcomes with
heads second and exactly 2
consecutive heads.
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
3. Use the definition of independence to
determine if the events are independent in
each specified pair.
P ( A Ç B ) = P ( A) · P ( B )
1 1 1
= ·
4 2 2
A and B are independent.
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
P ( A Ç C ) = P ( A ) · P (C )
1 1 1
= ·
8 2 4
A and C are independent.
P ( B Ç C ) = P ( B ) · P (C )
1 1 1
¹ ·
4 2 4
B and C are dependent.
✔
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
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7.1.3: Understanding Independent Events
Guided Practice
Example 3
Landen owns a delicatessen. He collected data on sales
of his most popular sandwiches for one week and
recorded it in the table below.
Bread
choice
Sandwich choice
Landen’s
club
Turkey
melt
Roasted
chicken
Veggie
delight
Country white
44
25
25
8
Whole wheat
24
28
26
34
Sourdough
24
27
24
31
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7.1.3: Understanding Independent Events
Guided Practice
Example 3, continued
Each of the following statements describes a pair of
events. For each statement, determine if the events
seem to be independent based on the data in the table.
A random customer orders Landen’s club sandwich
on country white bread.
A random customer orders the roasted chicken
sandwich on whole wheat bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
1. Find the totals of all the categories.
Sandwich choice
Landen’s Turkey Roasted
club
melt
chicken
Country white
44
25
25
Whole wheat
24
28
26
Sourdough
24
27
24
Total
92
80
75
Bread
choice
Veggie
delight
8
34
31
73
Total
102
112
106
320
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
2. For the first statement, assign variables as
names of the events.
LC: A random customer orders Landen’s club
sandwich.
CW: A random customer orders country white bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
3. Use the data to determine the relevant
probabilities.
P ( LC ) =
There were 320 sandwiches
sold, and 92 of them were
Landen’s club.
92
320
P (CW ) =
There were 320 sandwiches
sold, and 102 of them were on
country white bread.
102
320
P ( LC Ç CW ) =
44
320
There were 320 sandwiches
sold, and 44 of them were
Landen’s club on country white
bread.
7.1.3: Understanding Independent Events
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Guided Practice: Example 3, continued
4. Use the definition of independence to
determine if the events seem to be
independent.
P ( LC ÇCW ) = P ( LC ) · P (CW )
44
320
=
92
·
102
320 320
0.138 ¹ 0.092
Substitute probabilities
and simplify.
LC and CW seem to be
dependent, based on
the data.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
5. For the second statement, assign
variables as names of the events.
RC: A random customer orders a roasted chicken
sandwich.
WW: A random customer orders whole wheat bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
6. Use the data to determine the relevant
probabilities.
P ( RC ) =
There were 320 sandwiches
sold, and 75 of them were
roasted chicken.
75
320
P (WW ) =
There were 320 sandwiches
sold, and 112 of them were on
whole wheat bread.
112
320
P ( RC ÇWW ) =
26
320
There were 320 sandwiches
sold, and 26 of them were
roasted chicken on whole
wheat bread.
7.1.3: Understanding Independent Events
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Guided Practice: Example 3, continued
7. Use the definition of independence to
determine if the events seem to be
independent.
P ( RC ÇWW ) = P ( RC ) · P (WW )
26
320
=
75
·
112
320 320
Substitute probabilities
and simplify.
0.081» 0.082
RC and WW seem to be independent
based on the data.
✔
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
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7.1.3: Understanding Independent Events