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Conditional Probability
Multiplication Rule
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e
m
Outline
Conditional Probability
Multiplication Rule
Independence of Events
“Independent” versus “Mutually Exclusive”
The Birthday Problem
Simpson’s Paradox
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Xinghua Zheng
Probability II
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View Full Document Conditional Probability
Multiplication Rule
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Example, Stein’s class
±
What is the probability for a randomly selected student from
Professor Stein’s class this year to receive an A?
•
In the last 5 years Professor Stein has awarded 190 A’s out of
1000 students.
•
Can use
long run relative frequency
: the probability can be
estimated as 190
/
1000
=
0
.
19
±
Amy studies 10 hours or more every week for professor
Stein’s course. Amy is really interested in knowing “What’s the
probability for a student who studies 10 hours or more to be
awarded an A?"
•
Conditional probability
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Xinghua Zheng
Probability II
Conditional Probability
Multiplication Rule
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Conditional Probability
±
The probability of an event A, given that the event B has
occurred, is called the
conditional probability of A given B
•
Denoted by
P
(
A

B
)
±
Mathematically,
P
(
A

B
) :=
P
(
A
∩
B
)
P
(
B
)
,
•
Assuming
P
(
B
)
>
0
3 / 21
Xinghua Zheng
Probability II
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Multiplication Rule
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Interpretation
•
P
(
A

B
) =
P
(
A
∩
B
)
/
P
(
B
)
Geometrically, = the ratio of the area of the shaded part to
that of the right disk
•
P
(
B

A
) =
P
(
A
∩
B
)
/
P
(
A
)
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Xinghua Zheng
Probability II
Conditional Probability
Multiplication Rule
l
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Interpretation, ctd
The conditional probability
P
(
A

B
)
:
•
Restrict sample space to just event B
•
P
(
A

B
)
measures the chance of event A occurring in this new
sample space
•
In other words, if B occurs, then what is the chance of A
occurring
5 / 21
Xinghua Zheng
Probability II
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Multiplication Rule
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Example, Stein’s class, ctd
±
What’s the probability for a randomly selected student from Prof.
Stein’s class who studies 10 hours or more to be awarded an A?
•
Past data shows that 200 students worked 10 hours or more for
Stein’s course, and 120 of them were awarded A.
•
Deﬁne events A={to be awarded an A}, B = {studies 10 hours or
more}
•
To ﬁnd the answer to our question, we need to ﬁnd out
6 / 21
Xinghua Zheng
Probability II
Conditional Probability
Multiplication Rule
l
e
m
Example, Stein’s class, ctd
•
P
(
A

B
) =
P
(
A
∩
B
)
/
P
(
B
)
•
A
∩
B
={studies 10 hours or more
and
is to be awarded A}
•
By the longrun relative frequency,
P
(
A
∩
B
) =
120
/
1000
=
0
.
12,
P
(
B
) =
200
/
1000
=
0
.
2
•
Hence
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This note was uploaded on 12/28/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Fall '09 term at HKUST.
 Fall '09
 AnthonyChan

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