Probability Theory and Statistics - ProbabilityProbability can be used as measures of the degree of...

Probability Theory and Statistics

Probability

Dr.Ijaz Hussain

September 25, 2012

Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 1 / 11

Random Experiment

An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.

Random Experiment

An experiment whose results may not be predicted before occurrenceis known as random experiment.

Random Experiment

Examples

1 Tossing a coin

Random Experiment

Examples

1 Tossing a coin2 Tossing a well balanced dice

Random Experiment

Examples

1 Tossing a coin2 Tossing a well balanced dice3 Number of customer visiting a shop during one

Random Experiment

Examples

1 Tossing a coin2 Tossing a well balanced dice3 Number of customer visiting a shop during one4 Number accidents occurring at specific location during one month

Sample Space

A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.

Sample Space

For single coin the possible outcomes are 21 andS .S = {H,T}

Sample Space

For two coins the possible outcomes are 22 andS .S = {HH,HT ,TH,TT}

Sample Space

For three coins the possible outcomes are 23 andS .S = {HHH,HHT ,HTH,HTT ,THH,THT ,TTH,TTT}

Sample Space

For three coins the possible outcomes are 23 andS .S = {HHH,HHT ,HTH,HTT ,THH,THT ,TTH,TTT}

For a single dice the possible outcomes will be 61 andS .S = {1, 2, 3, 4, 5, 6}

For a pair of dice the possible outcomes will be 62

Tree Diagram

Basics of Sets

Universal set

Basics of Sets

Universal set

Union of sets

Basics of Sets

Universal set

Union of sets

Intersection of two sets

Basics of Sets

Universal set

Union of sets

Difference of two sets

Basics of Sets

Universal set

Union of sets

Complement of a set

Basics of Sets

Universal set

Union of sets

Complement of a set

Events

Event: It is subset of sample space defined on the basis ofcharacteristic of interest.

Events

Simple and compound Event: If there is only one outcome in anevent then its known as simple event otherwise it is called compoundevent.

Events

Equally likely Events/Outcomes If the two events/outcomes havesame chances of occurrence then they are said to be equally likely.

Events

Equally likely Events/Outcomes If the two events/outcomes havesame chances of occurrence then they are said to be equally likely.

Mutually Exclusive events The two events A and B are said to bemutually exclusive if they have nothing common outcomes.

Examples

Probability

Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.

Probability

Classical Method Probability is the ratio of number of favorableoutcomes to event A and number of all possible outcomes in a samplespace S .S if all outcomes are equally likely

Probability

P (A) = n(A)n(S .S)

Probability

P (A) = n(A)n(S .S)

0 ≤ P (A) ≤ 1

Probability

P (A) = n(A)n(S .S)

0 ≤ P (A) ≤ 1

The sum of the probabilities for all the experimental outcomes mustequal to 1

Relative Frequency Method

Probability

P (A) = n(A)n(S .S)

0 ≤ P (A) ≤ 1

The sum of the probabilities for all the experimental outcomes mustequal to 1

Relative Frequency Method

Subjective method

Counting Rules

Rule of Multiplication:If we have five suits, three ties and two pair of shoes then how manydifferent possible ways can be?

Counting Rules

Rule of Multiplication:If we have five suits, three ties and two pair of shoes then how manydifferent possible ways can be?For solution of above problem rule of multiplication may be used i.e5× 3× 2

Factorial:The product of first few natural numbers is called factorial,denoted as! and defined as;6! = 1× 2× 3× 4× 5× 6

Counting Rules

Rule of Multiplication:If we have five suits, three ties and two pair of shoes then how manydifferent possible ways can be?For solution of above problem rule of multiplication may be used i.e5× 3× 2

Factorial:The product of first few natural numbers is called factorial,denoted as! and defined as;6! = 1× 2× 3× 4× 5× 6

1 0! = 1, due to definition of gamma function2 n! = n (n − 1) (n − 2) (n − 3) (n − 4)!

Permutation and Combination

Permutation:The possible ways of selecting r objects out of n objects with respectto their order of selection is known as permutation.

1 nPr= n!

(n−r)!

Combination:The possible ways of selecting r objects out of n objects withoutrespect to their order of selection is known as combination.

1 nPr= n!

(n−r)!

Combination:The possible ways of selecting r objects out of n objects withoutrespect to their order of selection is known as combination.

1 nCr= n!

r !(n−r)!

Example 1

Suppose an experiment has five equally likely outcomes: E1, E2, E3, E4,E5. Assign probabilities to each outcome and show that the requirementsof probability are satisfied. What method did you use?

Example 1

Suppose an experiment has five equally likely outcomes: E1, E2, E3, E4,E5. Assign probabilities to each outcome and show that the requirementsof probability are satisfied. What method did you use?

Example 2

In the city of Milford, applications for zoning changes go through atwo-step process: a review by the planning commission and a finaldecision by the city council. At step 1 the planning commissionreviews the zoning change request and makes a positive or negativerecommendation concerning the change. At step 2 the city councilreviews the planning commissions recommendation and then votes toapprove or to disapprove the zoning change. Suppose the developerof an apartment complex submits an application for a zoning change.Consider the application process as an experiment.

1 How many sample points are there for this experiment? List the samplepoints.

Example 2

2 Construct a tree diagram for the experiment.

Example 2

2 Construct a tree diagram for the experiment.

Thanks

Probability Theory and Statistics - ProbabilityProbability can be used as measures of the degree of...

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