university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Probabilistic Operations Research Models
Paul Brooks Jill Hardin
Department of Statistical Sciences and Operations Research
Virginia Commonwealth University
BNFO 691 December 5, 2006
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Operations Research Models
Operations Research
Deterministic OR
Continuous
Variables
DiscreteVariables
Probabilistic OR
Discrete Time Continuous Time
Models
FunctionsLinear Linear Nonlinear Continuous
SpaceNonlinearFunctions Functions Functions
DiscreteSpace
ContinuousSpace
DiscreteSpace
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Linear Programming
Operations Research
Deterministic OR
Continuous
Variables
DiscreteVariables
Probabilistic OR
Discrete Time Continuous Time
Models
FunctionsLinear Nonlinear
FunctionsLinear
FunctionsNonlinearFunctions Space
Discrete ContinuousSpace
ContinuousSpaceSpace
Discrete
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Integer Programming
Operations Research
Deterministic OR
Continuous
Variables
DiscreteVariables
Probabilistic OR
Discrete Time Continuous Time
Models
FunctionsLinear Nonlinear
FunctionsLinear
FunctionsNonlinearFunctions Space
Discrete ContinuousSpace
ContinuousSpaceSpace
Discrete
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov Chains
Operations Research
Deterministic OR
Continuous
Variables
DiscreteVariables
Probabilistic OR
Discrete Time Continuous Time
Models
FunctionsLinear Nonlinear
FunctionsLinear
FunctionsNonlinearFunctions Space
Discrete ContinuousSpace
ContinuousSpaceSpace
Discrete
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Discrete-Event Simulation
Operations Research
Deterministic OR
Continuous
Variables
DiscreteVariables
Probabilistic OR
Discrete Time Continuous Time
Models
FunctionsLinear Nonlinear
FunctionsLinear
FunctionsNonlinearFunctions Space
Discrete ContinuousSpace
ContinuousSpaceSpace
Discrete
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Discrete vs. Continuous Models
Discrete
means “space between”
countable, e.g., integers, binary numbers
attributes, variables, time, space
Continuous
uncountable, e.g., real numbers, intervals of real numbers
attributes, variables, time, space
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Linear vs. Nonlinear Models
Linear
additivity - every function is the sum of the individualcontributions of activities
proportionality - the contribution of an activity to a functionis proportional to the level of the activity.
Hillier, FS and Lieberman, GJ. Introduction to Operations Research, 6th edition. McGraw-Hill, 1995.
Nonlinear
Additivity or proportionality (or both) are violated
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Probabilistic vs. Deterministic Models
ProbabilisticProbability is used to model behaviors that are uncertain orunknown
DeterministicRandomness is not considered; systems are assumed to betotally determined. Sensitivity analysis can help incorporateuncertainty into models.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
What is Probability?
Simply speaking probabilities are numbers between 0and 1 that reflect the chances of “something”happening.Synonymous with chance, likelihood, odds.Has different interpretations.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Cards and Dice
What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?
These are called events
Are you sure? What assumptions did you make?
Were they correct?
How can I correct these assumptions?
How can I determine a more accurate probability?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Classical/AnalyticalTheoretically determined probabilities
Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2
Advantagesprobabilities are accurateno experimentation requiredobjective
Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Classical/AnalyticalTheoretically determined probabilities
Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2
Advantagesprobabilities are accurateno experimentation requiredobjective
Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Classical/AnalyticalTheoretically determined probabilities
Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2
Advantagesprobabilities are accurateno experimentation requiredobjective
Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Classical/AnalyticalTheoretically determined probabilities
Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2
Advantagesprobabilities are accurateno experimentation requiredobjective
Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Classical/AnalyticalTheoretically determined probabilities
Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2
Advantagesprobabilities are accurateno experimentation requiredobjective
Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Relative Frequency/EmpiricalObserved proportion of successful events
10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.
Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective
Disadvantage: situation must be replicable
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Interpretations of Probability
Personal/Subjective“What do you think are the odds?”
What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?
Relies on expert information (definition of “expert” is fluid).Advantage:
always applicable - everybody has an opinionuseful in risk analysis
Disadvantage: difficult (sometimes impossible?) todetermine accuracy.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
1. Probability is always between 0 and 1.probability of an event E is written P(E)
2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.
3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.
4. The sum of the probabilities of all possible outcomes is 1.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
5. For two events A and B:P(A or B) = P(A) + P(B) − P(A and B)P(A and B) = P(A) + P(B) − P(A or B)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
6. For two events A and BP(B occurs given that A occurs)= P(B|A) = P(A and B)/P(A)⇒ P(A and B) = P(B|A)P(A)
If P(B|A) = P(B) and P(A|B) = P(A) then A and B are saidto be independent.
That is, knowing that one event will occur doesn’t give us anyinformation about the other.
If A and B are independent, then P(A and B) = P(A)P(B).
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
6. For two events A and BP(B occurs given that A occurs)= P(B|A) = P(A and B)/P(A)⇒ P(A and B) = P(B|A)P(A)
If P(B|A) = P(B) and P(A|B) = P(A) then A and B are saidto be independent.
That is, knowing that one event will occur doesn’t give us anyinformation about the other.
If A and B are independent, then P(A and B) = P(A)P(B).
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
6. For two events A and BP(B occurs given that A occurs)= P(B|A) = P(A and B)/P(A)⇒ P(A and B) = P(B|A)P(A)
If P(B|A) = P(B) and P(A|B) = P(A) then A and B are saidto be independent.
That is, knowing that one event will occur doesn’t give us anyinformation about the other.
If A and B are independent, then P(A and B) = P(A)P(B).
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
6. For two events A and BP(B occurs given that A occurs)= P(B|A) = P(A and B)/P(A)⇒ P(A and B) = P(B|A)P(A)
If P(B|A) = P(B) and P(A|B) = P(A) then A and B are saidto be independent.
That is, knowing that one event will occur doesn’t give us anyinformation about the other.
If A and B are independent, then P(A and B) = P(A)P(B).
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
DefinitionMutually exclusive events are non-overlapping— that is, theycannot happen at the same time.
A = randomly chosen person is maleB = randomly chosen person if femaleThese events are mutually exclusive.
A = randomly chosen person has blue eyesB = randomly chosen person has brown hairThese events are not mutually exclusive.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
Bayes’ RuleSuppose A1, A2, . . . , Ak are mutually exclusive events sothat
P(Ai) > 0, i = 1, . . . , kP(A1) + P(A2) + · · · + P(Ak ) = 1 (i.e., they are exhaustive).
Let B be another event with P(B) > 0. Then
P(Ai |B) =P(Ai and B)
P(B)=
P(B|Ai)P(Ai)k∑
i=1P(B|Ai)P(Ai)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
Bayes’ Rule
P(Ai |B) =P(Ai and B)
P(B)=
P(B|Ai)P(Ai)k∑
i=1P(B|Ai)P(Ai)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
Bayes’ Rule: ExampleSuppose that 5% of all athletes useperformance-enhancing drugs. Suppose further that forthe drug test in use, the false positive rate is 3% and thefalse negative rate is 7%.An athlete is tested, and her results are positive. What isthe probability that she uses drugs?
5%?97%something else?
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
Bayes’ Rule: ExampleWe want to find P(drug use|positive test). What we have are
P(positive test|no drug use) = P(false positive)P(negative test|drug use) = P(false negative)
Bayes’ Rule says that
P(drugs|positive) =P(positive|drugs)P(drugs)
P(positive|drugs)P(drugs) + P(positive|no drugs)P(no drugs)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Rules
Bayes’ Rule: Example
P(drugs|positive) = P(positive|drugs)P(drugs)P(positive|drugs)P(drugs)+P(positive|no drugs)P(no drugs)
= (0.93)(0.05)(0.93)(0.05)+(0.03)(0.95)
= 0.62
When we first met the athlete, we thought the chance of herbeing a drug user was 5%. We were able to use Bayes’ Rule,along with the test results, to update our “expert” information.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Example: Blood Types
Suppose the distribution of blood types (and genotypes) in apopulation is as follows:
A 40% AA 20%AO 20%
B 12% BB 6%BO 6%
AB 5% AB 5%O 43% OO 43%
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Example: Blood Types
What is the probability of producing a child with type O blood ifyour genotype is
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Example: Blood Types
What is the probability of producing a child with type O blood ifyour genotype isAO?P(OOchild) =P(mate with AO or BO, you contribute O, mate contributes O) +P(mate with OO, you contribute O)= (0.26)(0.5)(0.5) + (0.43)(0.5)= 0.28
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Example: Blood Types
What is the probability of producing a child with type O blood ifyour genotype isOO?P(OOchild) =P(mate with AO or BO, and mate contributes O) + P(mate with OO)= (0.26)(0.5) + 0.43= 0.56
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Example: Blood Types
What is the probability of producing a child with type O blood ifyour genotype isBB?P(OOchild) is zero! (impossible event)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Random Variables
Sometimes we’re more interested in some function ofan outcome, rather than the outcome itself.
If I flip a coin 5 times, how many are heads? This is afunction of the outcomes on five separate flips.How long will it be before Jill and Paul stop talking?
This function of the outcome is called a randomvariable.Observed value is determined by chance.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Random Variables
Need to know what values are possible— discrete orcontinuous?
Need to know what values are probable— how likely areeach of these values?
Probabilities defined by the probability density function (orprobability mass function)
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
pmf’s and pdf’s
Probability Mass Function
For a discrete random variable X , f (x) = P(X = x) for eachpossible value of x .
Probability Density Function
f (x) ≥ 0 for all x
P(a ≤ X ≤ b) = area under f (x) between a and b
Total area under f is 1
P(X = x) = 0
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
Probability Density Functions
Uniform
Normal
Triangular
Exponential
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Markov Property
Definition
A stochastic process with state variable Xt is said to possess theMarkov Property if
P(Xt+1 = it+1|X0 = i0, X1 = i1, . . . , Xt = it) = P(Xt+1 = it+1|Xt = it)
Translation: “The probabilities that a stochastic process movesto a new state depends only on the current state; theprobabilities are independent of all past events.”
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Markov Property
Definition
A stochastic process with state variable Xt is said to possess theMarkov Property if
P(Xt+1 = it+1|X0 = i0, X1 = i1, . . . , Xt = it) = P(Xt+1 = it+1|Xt = it)
Translation: “The probabilities that a stochastic process movesto a new state depends only on the current state; theprobabilities are independent of all past events.”
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Illustrations of the Markov Property
Let Xt = current position on the board after t rolls
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Illustrations of the Markov Property
Let Xt = location of a unit of ingested lead at time t
Tissue Bone
Bodyof
Outside
(Outside of Body)
Disposed
Blood
based on a model in Langkamp, G and Hull, J. Quantitative Reasoning and the Environment. Pearson, 2006.Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Illustrations of the Markov Property
Let Xt = location of a unit of ingested lead at time t
Tissue Bone
Bodyof
Outside
(Outside of Body)
Disposed
Blood
based on a model in Langkamp, G and Hull, J. Quantitative Reasoning and the Environment. Pearson, 2006.Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Illustrations of the Markov Property
Let Xt = political party of the U.S. Representative fromVirginia’s 3rd district after election t
Republican
DemocraticThirdParty
Party
Party
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Transition Matrix
ChildAA AB AO BB BO OO
AA 0.33 0.12 0.55 0 0 0AB 0.16 0.22 0.28 0.06 0.28 0AO 0.16 0.06 0.44 0 0.06 0.28
Par
ent
BB 0 0.33 0 0.12 0.55 0BO 0 0.16 0.16 0.06 0.34 0.28OO 0 0 0.33 0 0.12 0.55
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Blood Types
Let Xt = the blood type of a person in generation t
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Blood Types
Let Xt = the blood type of a person in generation t
AB
AA OO
BO
BBAO
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Transition Probabilities
Theorem
Let P be the transition matrix of a Markov Chain. ThenP(Xt = j |X0 = i) is the ij th entry of P t .
Corollary
The probability that a grandchild has genotype BB given thatthe grandparent has genotype AA is given in the appropriateentry of P2 (in our model).
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
Markov PropertyBlood Types II
Steady-State Behavior
Theorem
If all states are accessible from one another, then
limt→∞
P(Xt = j |X0 = i) = πj
where πj is the j th element of the vector π such that π = πP and∑
jπj = 1
This theorem gives us a means to calculate the steady statedistribution of genotypes. The quantity πj represents theprobability that, after a long time, a descendant has genotype j .It also represents the proportion of descendants that havegenotype j .
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
A Flowchart for Simulation Modeling
State objective
Output Data
Yes
Noand design study
a Model
Modelis
Valid?
Analyze
Collect
Data Experiments
Design
Construct/Program
Law and Kelton, Simulation Modeling and Analysis, 3rd Ed., 2000.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Probabilistic Modeling: Simulation vs. Analysis
Analysis
Advantage: Produces exact values for the characteristics of amodel given varied input. These values can easily be comparedfor determining optimal input values.
Disadvantage: Often requires strict assumptions about thenature of the model for any hope of solving for exact values.
Simulation
Advantage: Flexible in terms of assumptions required for model.
Disadvantage: Produces estimates of characteristics of a model;simulations need to be run for a wide variety of inputs and formany replications in order to derive reasonable estimates.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Types of Simulation
DefinitionA discrete-event simulation is a continuous-time,discrete-space simulation.
Definition
A Monte Carlo simulation is a static simulation.
Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Pitfalls of Simulation
Failure to have a well-definedset of objectives at thebeginning of the simulationstudy
Inappropriate level of modeldetail
Failure to collect good systemdata
Obliviously using simulationsoftware whose complexmacro statements may not bewell documented and may notimplement the desiredmodeling logic
Belief that easy-to-usesimulation packages require alower level of technicalcompetence
Failure to account correctly forsources of randomness in theactual system
Using arbitrary distributions asinput
Making a single replication of aparticular system design
Comparing alternative systemdesigns on the basis of onereplication
Law and Kelton, Simulation Modeling and Analysis, 3rd Ed., 2000.Paul Brooks, Jill Hardin
university-logo
Operations Research ModelsAxioms of Probability
Markov ChainsSimulation
The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Outline
1 Operations Research Models2 Axioms of Probability
DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables
3 Markov ChainsMarkov PropertyBlood Types II
4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation
Paul Brooks, Jill Hardin