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Statistics for Economist
Ch.9 Probability
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Chances, Probabilities
Marbles in Boxes: drawing with or without replacement
Listing the Ways
Venn Diagram & Exclusive Events
The Addition Rule
Conditional Probabilities
The Multiplication Rule
Partition & Bayes’ Theorem
Independence
Exclusiveness & Independence, Addition & Multiplication
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INDEX
STATISTICS
1
Chances, Probabilities
2
Marbles in Boxes
: drawing with or without replacement
3
Listing the Ways
4
Venn Diagram & Exclusive Events
5
The Addition Rule
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1. Chances, Probabilities
STATISTICS
Probabilities?
 Frequentist View
The Chances of something gives the percentage of
time it is expected to happen, when the basic process
is done over and over again, independently and under
the same conditions.
 Subjective View
Typically in cases where repeated trial is impossible.
Subjective representation of how likely it is. Defined
regardless of repetition.
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1. Chances, Probabilities
STATISTICS
Properties of Probability
 Probabilities are between 0% and 100%.
 If the probability that an event A will occur is P(A),
the probability that the event A will not occur is P(A).
“The event A will not occur” is also considered another
event, which is called “complementary event AC of A”
P( A ) 1  P( A)
C
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INDEX
STATISTICS
1
Chances, Probabilities
2
Marbles in Boxes
: drawing with or without replacement
3
Listing the Ways
4
Venn Diagram & Exclusive Events
5
The Addition Rule
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STATISTICS
2. Marbles in Boxes :
drawing with or
without replacement
Marbles in Boxes
 The probability of drawing red marbles from A or B?
Box A (red marbles 3, blue marble 2)
Box B (red marbles 30, blue marble 20)
Number of Red Marbles
Total Number of Marbles
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STATISTICS
2. Marbles in Boxes :
drawing with or
without replacement
Drawing with or without replacement
1st
1
trial
2
3
Suppose that the outcome of the 1st trial is card 3
2nd
trial
The outcome of the 2nd trial is depend upon whether drawing
with or without replacement
1
2
3
1
drawing WITH replacement
2
Drawing WITHOUT replacement
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INDEX
STATISTICS
1
Chances, Probabilities
2
Marbles in Boxes
: drawing with or without replacement
3
Listing the Ways
4
Venn Diagram & Exclusive Events
5
The Addition Rule
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STATISTICS
3. Listing the Ways
Possible Ways
Throwing a Pair of Dice
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3. Listing the Ways
STATISTICS
Combinations with a total of 9,10 rolled with three dice
9 : (1,2,6), (1,3,5), (1,4,4), (2,5,5), (2,3,4), (3,3,3)
10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4)
Same in number of combinations → Same in total possible ways? No!
Total of 9
# of combs.
Total of 10
# of combs.
1,2,6
6
1,3,6
6
1,3,5
6
1,4,5
6
1,4,4
3
2,2,6
3
2,2,5
3
2,3,5
6
2,3,4
6
2,4,4
3
3,3,3
1
3,3,4
3
Total
25
Total
27
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INDEX
STATISTICS
1
Chances, Probabilities
2
Marbles in Boxes
: drawing with or without replacement
3
Listing the Ways
4
Venn Diagram & Exclusive Events
5
The Addition Rule
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4. Venn Diagram & Exclusive Events
STATISTICS
Venn Diagram
Venn Diagram
A Venn diagram is a diagram using a rectangle and
some inner circles to represent one or more events
A
B
A
B
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4. Venn Diagram & Exclusive Events
STATISTICS
Disjoint
Events
If some two events cannot come together,
the two events are called ‘Exclusive Events’ or
‘Mutually Exclusive.’
A
1,3,5
B
A
6
5
(a) Exclusive Events
B
1
3
2
(b) Not Exclusive
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INDEX
STATISTICS
1
Chances, Probabilities
2
Marbles in Boxes
: drawing with or without replacement
3
Listing the Ways
4
Venn Diagram & Exclusive Events
5
The Addition Rule
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STATISTICS
5. The Addition Rule
Addition Rule
 P(A or B):
the Probability that at least one event will occur
among the two
 P(A and B):
the probability that the two events will come together
If they are mutually exclusive, the Probability is 0.
 Generalized Addition Rule:
P(A or B)=P(A) + P(B) - P(A and B)
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INDEX
STATISTICS
6
Conditional Probabilities
7
The Multiplication Rule
8
Partition & Bayes’ Theorem
9
Independence
10
Exclusiveness & Independence,
Addition & Multiplication
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6. Conditional Probabilities
STATISTICS
Conditional Probability
Ex) A deck of cards is shuffled and the top two cards are
put on a table, face down. You win \1,000 if the
second card is Q of hearts.
a) What is the probability of winning the won?
b) You turn over the first card. It is the seven of clubs.
Now what is the probability of winning?
a) Non-conditional probability  Pr(the 2nd card is Q of hearts)
☞ 1/52
b) Conditional probability  Pr (2nd card is Q of hearts | 1st card is 7 of clubs)
☞ 1/51
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INDEX
STATISTICS
6
Conditional Probabilities
7
The Multiplication Rule
8
Partition & Bayes’ Theorem
9
Independence
10
Exclusiveness & Independence,
Addition & Multiplication
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7. The Multiplication Rule
STATISTICS
Multiplication Rule
 Joint Probability
P(A and B), the probability the two will come together
 Conditional Probability
P(A|B), the probability that event A will occur given the
occurrence of event B
 Mmarginal Probability
P(A) or P(B), non-conditional probability
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7. The Multiplication Rule
STATISTICS
Multiplication Rule
 Narrow Meaning:
When some two events are mutually independent,
the probability that the two will come together is
acquired by multiplying each non-conditional
probability.
P(A and B)=P(A)·P(B)
 Generalized Multiplication Rule:
The probability that both of two events will occur is acquired by
multiplying the probability of one event’s occurrence and the
conditional probability of another event’s occurrence given the
occurrence of the event.
P(A and B)=P(A)·P(B|A)
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INDEX
STATISTICS
6
Conditional Probability
7
The Multiplication Rule
8
Partition & Bayes’ Therem
9
Independence
10
Exclusiveness & Independence,
Addition & Multiplication
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8. Partition & Bayes’ Theorem
STATISTICS
Concept
 Partition
A division of a set into Collectively Exhaustive and
Mutually Exclusive events
Ex) when a die is rolled, the event of even numbers and
the event of odd numbers make up a partition
Counter Ex) Event of odd numbers and Event of 6.
Event of odd numbers and Event of numbers larger than 2.
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8. Partition & Bayes’ Theorem
STATISTICS
Partition of Union & Partition of B
Partition of Union
S
AC
A
=
+
Partition of B
A
B
AC and B
A and B
=
+
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8. Partition & Bayes’ Theorem
STATISTICS
Conditional Probability
 Conditional Probability P(A|B)
Probability that event A will occur given the occurrence of
event B.
Relative magnitude of event (A & B) compared with event B
the Convex Lens
P( A | B) =
=
=
Circle of Right Side
+
P( A and B)
P( A and B)
P( A | B) 

P( B)
P( A and B)  P( AC and B)
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STATISTICS
8. Partition & Bayes’ Theorem
Tree Diagram
B given A
A
Two routes to event B:
collectively exhaustive & mutually exclusive
AC
P( A B) 
B given AC
P(Above &  )
P( A and B)

P(Above &  )  P(Below & ) P( A and B)  P( AC and B)
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8. Partition & Bayes’ Theorem
STATISTICS
Bayes Theorem (1)
Simple form : if P (B) > 0
P(A|B) 
P(A and B)
P(A and B)
P(A)P(B|A)


P(B)
P(A and B)  P(AC and B) P(A)P(B|A)  P(AC )P(B|AC )
Q) If one selected the right answer to the multiple choice
question having 4 possible answers (event B),
The probability that one selected it knowing surely (event A)?
Prior Probability : P (A)=1/2
Posterior Probability : P (A|B)=4/5
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8. Partition and Bayes’ Theorem
STATISTICS
an example of partition
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8. Partition and Bayes’ Theorem
STATISTICS
Bayes’ Theorem (2)
Generalized form
if P (B) > 0 ,
Let A1 , A2 ,    , Am form a partition for S
P( A1 and B)
P( A1 ) P(B | A1 )
P( A1 | B) 

P( B)
P( A1 ) P(B | A1 )    P ( Am )P(B | Am )
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INDEX
STATISTICS
6
Conditional Probability
7
The Multiplication Rule
8
Partition & Bayes’ Therem
9
Independence
10
Exclusiveness & Independence,
Addition & Multiplication
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STATISTICS
9. Independence
Independence & Dependence
 If the PROBABILITY that the other event occur is not changed
whether one event occur or not, we call the two events are
‘independent’. Otherwise, we call them ‘dependent’
 If event A and event B are independent,
P(A|B)=P(A)
P(B|A)=P(B)
 Narrow Meaning of Multiplication Rule
: P(A and B) = P(A) P(B)
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INDEX
STATISTICS
6
Conditional Probability
7
The Multiplication Rule
8
Partition & Bayes’ Therem
9
Independence
10
Exclusiveness & Independence,
Addition & Multiplication
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STATISTICS
10. Exclusiveness & Independence,
Addition & Multiplication
Mutual Exclusiveness & Mutual Independence
 Mutual Exclusiveness
if one event occurs then the other cannot occur
 Mutual Independence
If the probability that the other event occur is not changed whether
one event occur or not
 Mutually Exclusive events are Mutually Dependent
If events A, B are mutually exclusive and event A has occurred, the
probability that event B occurs becomes 0
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STATISTICS
10. Exclusiveness & Independence,
Addition & Multiplication
Addition Rule & Multiplication Rule
 Addition Rule
Regarding the probability that at least one event will occur,
Addition rule of narrow meaning is possible only when the events
are mutually exclusive. (otherwise, one should subtract the
overlapped part)
 Multiplication Rule
Regarding the probability that the two events come together,
Multiplication rule of narrow meaning is possible only when the
events are mutually independent.
(otherwise, one should multiply the marginal probability of one
event and the conditional probability of the other event.)
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