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Probability Theory and Statistics
Probability
Dr.Ijaz Hussain
CIIT
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.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Random Experiment
An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.
An experiment whose results may not be predicted before occurrenceis known as random experiment.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Random Experiment
An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.
An experiment whose results may not be predicted before occurrenceis known as random experiment.
Examples
1 Tossing a coin
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Random Experiment
An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.
An experiment whose results may not be predicted before occurrenceis known as random experiment.
Examples
1 Tossing a coin2 Tossing a well balanced dice
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Random Experiment
An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.
An experiment whose results may not be predicted before occurrenceis known as random experiment.
Examples
1 Tossing a coin2 Tossing a well balanced dice3 Number of customer visiting a shop during one
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Random Experiment
An experiment which produces different results even though it isrepeated large number of times under same conditions is calledrandom experiment.
An experiment whose results may not be predicted before occurrenceis known as 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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 2 / 11
Sample Space
A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 3 / 11
Sample Space
A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.
For single coin the possible outcomes are 21 andS .S = {H,T}
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 3 / 11
Sample Space
A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.
For single coin the possible outcomes are 21 andS .S = {H,T}
For two coins the possible outcomes are 22 andS .S = {HH,HT ,TH,TT}
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 3 / 11
Sample Space
A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.
For single coin the possible outcomes are 21 andS .S = {H,T}
For two coins the possible outcomes are 22 andS .S = {HH,HT ,TH,TT}
For three coins the possible outcomes are 23 andS .S = {HHH,HHT ,HTH,HTT ,THH,THT ,TTH,TTT}
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 3 / 11
Sample Space
A list of all possible outcomes of random experiment is known assample space denoted by S.S or S.
For single coin the possible outcomes are 21 andS .S = {H,T}
For two coins the possible outcomes are 22 andS .S = {HH,HT ,TH,TT}
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 3 / 11
Basics of Sets
Universal set
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Basics of Sets
Universal set
Union of sets
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Basics of Sets
Universal set
Union of sets
Intersection of two sets
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Basics of Sets
Universal set
Union of sets
Intersection of two sets
Difference of two sets
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Basics of Sets
Universal set
Union of sets
Intersection of two sets
Difference of two sets
Complement of a set
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Basics of Sets
Universal set
Union of sets
Intersection of two sets
Difference of two sets
Complement of a set
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 4 / 11
Events
Event: It is subset of sample space defined on the basis ofcharacteristic of interest.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 5 / 11
Events
Event: It is subset of sample space defined on the basis ofcharacteristic of interest.
Simple and compound Event: If there is only one outcome in anevent then its known as simple event otherwise it is called compoundevent.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 5 / 11
Events
Event: It is subset of sample space defined on the basis ofcharacteristic of interest.
Simple and compound Event: If there is only one outcome in anevent then its known as simple event otherwise it is called compoundevent.
Equally likely Events/Outcomes If the two events/outcomes havesame chances of occurrence then they are said to be equally likely.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 5 / 11
Events
Event: It is subset of sample space defined on the basis ofcharacteristic of interest.
Simple and compound Event: If there is only one outcome in anevent then its known as simple event otherwise it is called compoundevent.
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 5 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
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
P (A) = n(A)n(S .S)
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
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
P (A) = n(A)n(S .S)
0 ≤ P (A) ≤ 1
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
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
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Probability
Probability can be used as measures of the degree of uncertaintyassociated with the event of interest.
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
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 6 / 11
Counting Rules
Rule of Multiplication:If we have five suits, three ties and two pair of shoes then how manydifferent possible ways can be?
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 7 / 11
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
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 7 / 11
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)!
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 7 / 11
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.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 8 / 11
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.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 8 / 11
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 nCr= n!
r !(n−r)!
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 8 / 11
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?
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 9 / 11
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?
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 9 / 11
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.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 10 / 11
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.
2 Construct a tree diagram for the experiment.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 10 / 11
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.
2 Construct a tree diagram for the experiment.
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 10 / 11
Thanks
Dr.Ijaz Hussain (CIIT) Probability Theory and Statistics September 25, 2012 11 / 11