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STATISTICS
Basic Concept of Probability
Ir. Mahmud Sudibandriyo MSc., PhD
September 24, 2008
1
Topics
¾ Concepts and Definitions
¾ Compound Event Probabilities
¾ Enumeration Technique
September 24, 2008
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Concepts and Definitions
Probability Experiment: activities where an outcome, response, or
measurement is obtained.
Sample space (S): A set of values which covers all possibility of
outcome, response, or measurement.
Event : A subset of all outcome, response, or measurement in the
sample space.
Example: we inspect 3 pumps one by one in sequence, we note G
for a pump in a good condition and F for a fail pump..
S
S = {GGG,GGF,GFG,FGG,GFF,FGF,FFG,FFF}
A
If A is an event where we find one fail pump, then:
A = {GGF, GFG, FGG}
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Probability:
Probability = 1 if the event is definitely occurs
Probability = 0 if the event is impossible.
Classical Definition: P(A) = fA/N
and P(Ā) = 1- P(A)
exp: probability to get Ace in bridge cards is 4/52
probability to get non Ace is 1 – 4/52 = 48/52
Relative Frequency Definition: P (A) = lim fA/N
NÆ∞
Subjective Definition: based on expert judgment
exp: what is the probability of PERSIJA to win the game
against PSIS ?
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Compound Event Probabilities
Compound Event: a combination of 2 or more simple events
Conditional probability: P (A|B) = P (A∩B)/ P(B) ; P(B) >0
exp: In PC production, 60% is installed word processor (A), 40%
is installed with spreadsheet (B), 30% is installed with both
of them. If one buy a PC with spreadsheet in it, what the
probability this computer is also installed with word
processor ?
P(A) = 0.6 ; P(B) = 0.4 ; P (A∩B) = 0.3 ; P(A|B)=0.3/0.4=0.75
B
A
0.3
September 24, 2008
0.3
0.1
A∩B
0.1
0.3
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Independent and Dependent Event
Independent event : P(A|B) = P(A) or P(B|A) = P(B)
exp: A: “heads on the fifth toss”, B: “heads on the sixth toss”
Dependent event : exp. conditional probability
Mutually Exclusive
P(A|B) = 0 and P(B|A) = 0
exp: A: “drawing an ace from a deck of cards”
B: “drawing a King”
since both ace and king cannot be drawn in a single
draw, they are thus mutually exclusive
A
September 24, 2008
B
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Laws of Probabilities in Compound Event
Multiplication Law
Independent Event
P(A and B and C and…) = P(A∩B∩C∩…)= P(A).P(B).P(C)…
exp: Probability of heads on both the fifth and sixth tosses
P(A).P(B) = ½ . ½ = ¼
Dependent Event
P(A and B) = P(A∩B) = P(A|B).P(B) = P(B|A).P(A)
exp: see Ref. Book p. 51
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Addition Law
P(A or B) = P(AUB) = P(A)+P(B) – P(A∩B)
P(A or B or C) = P(AUBUC) = P(A)+P(B)+P(C)-P(A∩B)
-P(A∩C) – P(B∩C) + P(A∩B∩C)
Exp:
see Ref. book p. 52-53
Bayes Formulation
A combination of conditional probability and multiplication law
Exp: see Ref. Book p 55
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Enumeration Technique
Probability Tree
A1
A2
P(A1)
P(A2)
B1
P(B1|A1)
B2
P(B2|A1)
B3
P(B3|A1)
B1
P(B1|A2)
B2
P(B2|A2)
B3
P(B3|A2)
P(A1∩B1)
P(A1∩B2)
P(A1∩B3)
P(A2∩B1)
P(A2∩B2)
P(A2∩B3)
Exp. See textbook p. 56-57
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Combinatorial Analysis
Permutation
An arrangement of r out of n
objects with attention given to
the order of arrangement
n!
n Pr = P ( n, r ) = Pn , r = P =
(n − r )!
n
r
Exp: the number of permutation
of the letters a,b,c taken two at
a time is 3P2 = 3.2/1 = 6
Combination
A selection of r out of n objects
with no attention given
to the
n!
n Pr
=
n Cr =
r!(n − r )! r!
order of arrangement
Exp: 3C2 = 6/2 = 3
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