1 Definitions Experiment – a process by which an observation ( or measurement ) is observed Sample Space (S)- The set of all possible outcomes (or results)

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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: