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PROBABILITY
References:
Budiyono. 2010. Statistika untuk Penelitian: Edisi Kedua.
Surakarta: UNS Press.
Spigel, M. R. 1882. Probability and Statistics. Singapore:
McGraws-Hill International Book Company.
Walpole, R. E. 1982. Introduction to Statistics. New York:
Macmillan Publishing Co.,Inc.
EXPERIMENTS, SAMPLE
SPACES, AND EVENTS
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In statistics, we have what we call statistical
experiments, or experiments, for brieftly
Sample space is a set of all possible
outcome of an experiment.
Sample space is denoted by S
The member of sample space is called
sample point.
An event is a subset of a sample space.
Theorem
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Theorem 1
On a finite sample space S, having n
elements, the number of events on S is
2n(S).
Experiment of tossing a coin
once
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S = {A, G}
or S = {H, T}
n(S) = 2
The possible events on S are:
1. E1 = ;
2. E2 = {A};
3. E3 = {G}; and
4. E4 = {A, G}
The number of events on S is 22
Experiment of tossing a die
once
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S = {1, 2, 3, 4, 5, 6}
n(S) = 6
The example of events on S are:
1. E1 = ;
2. E2 = {1, 2, 3, 4, 5, 6};
3. E3 = {1, 3, 5}; and
4. E4 = {1, 2, 4}, etc
The number of events on S is 26
Example of tossing 3 coins
once (or 1 coin three times)
Example of tossing 3 coins
once (or 1 coin three times)
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S = {AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG}
or S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
n(S) = 8
The number of events on S is 28
An example of events on S are:
1. E1 = {}
2. E2 = S
3. E3 = {AAA}
4. E4 = {AAG, AGA, GAA}, etc
Experiment of tossing 2 dice
once (or 1 dice two times)
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S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1),
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), ...
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)}
n(S) = 36
An example of events on S are:
1. E1 = ;
2. E2 = {(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)};
3. E3 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}; and
4. E4 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, etc
The number of events on S is 236
Experiment of tossing a die
and a coin once
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S = {(1,A), (1,G), (2,A), (2,G), (3,A), (3,G),
(4,A), (4,G), (5,A), (5,G), (6,A), (6,G)}
n(S) = 12
An example of events on S are:
1.
2.
3.
4.
5.
E1 = ,
E2 = S
E3 = {(1,A), (2,A), (3,A), (4,A), (5,A), (6,A)}
E2 = {(2,A), (2,G)}
E3 = {(4,A)}, etc
The Definition of Probability
If an event A assosiated with an experiment having
sample space S, in which all possible sample points
have a same probability to occur, then the probobility
of event A, denoted by P(A), is defined by the
following formulae:
n ( A)
P( A) 
n (S)
The definition is called classic definition or a priori
definition of a probability.
Example 1
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Experiment of tossing a die once
S = {1, 2, 3, 4, 5, 6}; n(S) = 6
The example of events on S are:
1. E1 = ;
2. E2 = {1, 2, 3, 4, 5, 6};
3. E3 = {1, 3, 5}; and
4. E4 = {1, 2, 4}, etc
P(E1 ) 
n (E1 ) 0
 0
n (S)
6
P( E 2 ) 
P( E 3 ) 
n (E 3 ) 3
  0.5
n (S)
6
P( E 4 ) 
n (E 2 ) 6
 1
n (S)
6
n (E 4 )
 3  0.5
n (S)
6
Example 2
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Experiment of tossing 2 dice once (or 1 die two
times)
S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1),
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), ...
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)}
a. Find the probability of getting the same number!
E1 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
n(E1) = 6
P(E1) 
n (E1) 6 1


n (S) 36 6
Example 3
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Experiment of tossing 2 dice once (or 1 dice two
times)
S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1),
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), ...
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)}
b. Find the probability of scoring a total of 7 points
E2 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E2) = 6
P( E 2 ) 
n (E 2 ) 6 1


n (S)
36 6
Example 4
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Experiment of tossing 2 dice once (or 1 dice two
times)
S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1),
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), ...
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)}
c. Find the probability of getting even score on the
first die.
E3 = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),}
Example 5
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A marble is drawn at random from a
box containing 10 red, 30 white, 20
blue, and 15 orange marbles.
Find the probability that it is orange or
red
Example 6
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A card is drawn at random from an
ordinary deck of 52 playing cards.
Find the probability that it is an ace
Example 7
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A card is drawn at random from an ordinary
deck of 52 playing cards.
Find the probability that it is a jack of hearts
Example 8
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A card is drawn at random from an ordinary
deck of 52 playing cards.
Find the probability that it is a three of clubs
or a six of diamonds
The Axioms of Probability
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Postulate 1
P(A) is a real non-negatif number for every A in
sample space S, i.e. P(A) ≥ 0 for every event A.
Postulate 2
P(S) = 1 for every sample space S.
Postulate 3
If A1, A2, A3 , ... are mutually exclusive events in
sample space S, then:
P(A1A2A3 ...) = P(A1) + P(A2) + P(A3) + ...
Theorems:
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Theorem 2
if A is an event on S and Ac is the complement of A,
then:
P(Ac) = 1  P(A)
Theorem 3
For every event A on S,
0  P(A)  1
Theorem 4
The probability of an empty set is zero, i.e. :
P() = 0
Theorems:
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Theorem 5
For every event A and B on S,
P(A  B) = P(A) + P(B)  P(A  B)
Theorem 6
For every event A, B, and C on S,
P(ABC) = P(A) + P(B) + P(C)  P(AB)  P(AC)
 P(BC) + P(ABC)
Theorem 7
If A and B are events on S and A  B, then:
P(A)  P(B)
Example
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Suppose P(A) = 0.5, P(B) = 0.4, and P(AB)
= 0.3. Find:
a. P(AB)
b. P(ABc)
c. P(AcBc)
d. P(Ac Bc)
Example
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In a class in wich there is 40 students on it, is given
the following data. As many as 25 students like
football, as many as 15 students like tennis, and as
many as 5 students like both sports. A person is
called randomly. What is the probability that someone
who called likes:
a. football
b. football or tennis
c. football and tennis
Disjoint Events
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Event A and B on S are called disjoint
events (kejadian saling asing, saling
lepas) if:
AB=
Conditional Probability
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If A and B are two events on S in which
P(A)  0, then the conditional
probability of B given A, denoted by
P(B|A), is defined as follows:
P(A  B)
P( B | A ) 
P ( A)
Theorems on Conditional
Probability
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Theorem 8
For any two events A and B on a sample
space S, we have:
P (A  B) = P(A) P(B|A)
Theorem 9
For any events A1, A2, A3, ... on a a sample
space S, we have:
P(A1  A2  A3 ... ) = P(A1) P(A2|A1) P(A3|A1A2) ...
Total Probabity Theorem
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If events B1, B2, B3, ... , Bk form a
partition on S and P(Bi)  0 for every i
= 1, 2, 3, ... , k, then for any event A
on S in which P(A)  0, then:
P(A) = P(B1)P(A|B1) + P(B2)P(A|B2) +
... + P(Bk)P(A|Bk)
It can be written as:
k
P(A)   P(Bi )P(A | Bi )
i 1
Example
Bayes’ Theorem
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Suppose events B1, B2, B3, ... , Bk form a
partition on a sample space S and P(Bi)  0
for every i = 1, 2, 3, ... , k. Then for any
event A on S with P(A)  0, we have:
P(B r | A) 
P( B r ) P( A | B r )
k
 P ( Bi ) P ( A | Bi )
i 1
for every i = 1, 2, 3, ... , k
Example
Example
The box number I contains 3 red balls and 2 blue balls. The box
number II contains 2 red balls and 8 blue balls. A coin is tossed.
If the “angka” occurs, then any ball is drawn randomly from the
box number I. Otherwise, if the “gambar” occurs then any ball
is drawn randomly from the box number II.
a. What is the probability that the ball is drawn from the box
number I?
b. What is the probability that the ball is drawn from the box
number II?
c. What is the probability any red ball is drawn?
d. What is the probability any blue ball is drawn?
e. Someone who toss the coin does not know wether occurs
“angka” or “gambar”, but he knows that a red ball is drawn.
What is the probability that the ball is drawn from the box
number I?
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Independent Events
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Definition:
Any two events A and B are called
independent events if
P(A  B) = P(A)P(B)
Theorems:
If A and B are independent events, then:
1. A and Bc are independent events,
2. Ac and B are independent events, and
3. Ac and Bc are independent events
Example
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Three balls are drawn successively from a box
containing 6 red balls, 4 white balls, and 5 blue balls.
Find the probability that they are drawn in order red,
white, and blue, if the sampling is:
a. with replacement
b. without replacement
Example
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Given two independent events A and B
having P(A) = 0.5 and P(B) = 0.4. Find: