1 definitions experiment – a process by which an observation ( or measurement ) is observed sample...
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Definitions
Experiment – a process by which an observation
( or measurement ) is observed
Sample Space (S)- The set of all possible outcomes (or results) of an experiment
Event (E) – a collection of outcomes
SEei .
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Example
Experiment : Toss a balanced die once and observe its uppermost face
Sample Space =S={1,2,3,4,5,6}Events: 1.observe a even number E= { 2,4,6} 2. observe a number less than or
equal to 4 F= { 1,2,3,4}
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Probability
Given a event (E) , we would like to assign it a number, P(E)
P(E) is called the probability of E (likelihood that E will occur)
Practical Interpretation The fraction of times that E happens out of a huge
number of trials of the same experiment will be close to P(E)
1)(0 EP
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Types of Probabilities
Theoretical Empirical
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Theoretical Probabilities
Used if the outcomes of an experiment are equally likely to occur
If E is an Event
spacesampleinoutcomesofnumber
EeventinoutcomesofnumberEP )(
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Example
Toss a balanced die once and observe its uppermost face
S={1,2,3,4,5,6}
Let G=“observe a number divisible by 3”
G={3,6}
Then P(G)=2/6=1/3
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Empirical Probabilities
Used when theoretical probabilities cannot be used
The experiment is repeated large number of times
If E is an Event
trialsofnumber
happensEtimesofnumberEP )(
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Example
The freshman class at ABC college
- 770 students
- 485 identified themselves as “smokers”
Compute the empirical probability that a randomly selected freshman student from this class is not a smoker
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Example-contd.
E= event that a randomly chosen student from this class is not a smoker
P(E)= 285/770=0.37
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Properties I
1.
2. If E is certain to happen
3. If E and F cannot both happen
4.
1)(0 EP1)( EP
)()()( FPEPForEP
1)( SP
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Union
Def. The union of two sets, E and F, is the set of outcomes in E or F .
Example:
E= { 2,4,6}
F= { 1,2,3,4}
}6,4,3,2,1{FE
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Intersection
Def. The intersection of two sets, E and F, is the set of outcomes in E and F .
Example:
E= { 2,4,6}
F= { 1,2,3,4}
}4,2{FE
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Mutually Exclusive
Def. Two events, E and F, are mutually exclusive if they have no outcomes in common, i.e. .
If E and F are mutually exclusive, then
)()()( FPEPFEP
FE
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This property can be extended to more than two events.
For any two events, E and F,
)()()()( FEPFPEPFEP
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Complement of an Event
Def. The complement of an event, E, is the event that E does not happen .
Example: S={1,2,3,4,5,6} E= { 2,4,6}
Does E and have common outcomes?
}5,3,1{CECE
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Since the two events are Mutually Exclusive
)(1)(
)()(1
)()()(
)()()(
EPEP
EPEP
EPEPSP
EPEPEEP
C
C
C
CC
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2
12
11
)(1)(
EPEP C
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Assign probability to each outcome Each probability must be between 0 and 1 The sum of the probabilities must be equal to 1
If the outcomes of an experiment are all equally likely, then the probability of each outcome is given by ,where n is the number of possible outcomes n
1
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DeMorgan’s Laws
)(1))(()(
)(1))(()(
FEPFEPFEP
FEPFEPFEPCCC
CCC
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Project Focus
• How can probability help us with thedecision on whether or not to attempt a loan work out?
Events:
S- an attempted work out is successful
F- an attempted work out fails
Goal:
P(S) – Probability of S or fraction of past work out arrangements which were successful
P(F) - Probability of F or fraction of past work out arrangements which were unsuccessful?
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Using “Countif” function in Excel
Counts the number of cells within a given range that meets the given criteria
Fields for the function
1. Range
2. Criteria
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Number of Successes
Number of Failures
Fraction of Successes
Fraction of Failures P (S ) P (F )
3,818 4,408 0.464138099 0.535861901 0.464 0.536
Estimated ProbabilitiesCounting Fractions
Project Focus – Basic Probability
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More on Events S & F
F is the complement of S
)S(P)F(P
)S(P)S(P C
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Recall: