What Is Probability?What Is Probability?

Farrokh Alemi Ph.D.Farrokh Alemi Ph.D.Professor of Health Administration and PolicyProfessor of Health Administration and Policy

College of Health and Human Services, George Mason UniversityCollege of Health and Human Services, George Mason University4400 University Drive, Fairfax, Virginia 220304400 University Drive, Fairfax, Virginia 22030

703 993 1929 703 993 1929 falemi@gmu.edufalemi@gmu.edu

Lecture OutlineLecture Outline

1.1. What is probability?What is probability?

2.2. Assessment of rare probabilitiesAssessment of rare probabilities

3.3. Calculus of probabilityCalculus of probability

4.4. Conditional independence Conditional independence

5.5. Causal modelingCausal modeling

6.6. Case based learningCase based learning

7.7. Validation of risk modelsValidation of risk models

8.8. Examples Examples

Lecture OutlineLecture Outline

1.1. What is probability?What is probability? PartitioningPartitioning Probability axiomsProbability axioms Subjective probabilitySubjective probability Hazard functions and related termsHazard functions and related terms

2.2. Assessment of rare probabilitiesAssessment of rare probabilities3.3. Calculus of probabilityCalculus of probability4.4. Conditional independence Conditional independence 5.5. Causal modelingCausal modeling6.6. Case based learningCase based learning7.7. Validation of risk modelsValidation of risk models8.8. Examples Examples This is a new

way of thinking.

Why measure uncertainty?Why measure uncertainty?

To make tradeoffs among To make tradeoffs among uncertain events  uncertain events 

Measure combined effect of Measure combined effect of several uncertain eventsseveral uncertain events

To communicate about To communicate about uncertaintyuncertainty

DefinitionDefinition

Probability quantifies how Probability quantifies how uncertain we are about uncertain we are about future eventsfuture events

More Precise DefinitionMore Precise Definition

A probability function assigns A probability function assigns numbers to events in a sample numbers to events in a sample space so that:space so that:

1.1. At least one event from the possible At least one event from the possible sample must happen.sample must happen.

2.2. Probability of any event is greater than Probability of any event is greater than or equal to zero.or equal to zero.

3.3. Probability of a complement of an Probability of a complement of an event is one minus the probability of event is one minus the probability of the eventthe event

4.4. Probability of two mutually exclusive Probability of two mutually exclusive event occurring is the sum of eachevent occurring is the sum of each

What is probability?What is probability?

What is Probability?What is Probability?

A

P(A)=

A

P(A)=

What is probability?What is probability?

A

P(A)=

A

P(A)=

All non A events

P( not A ) =

DefinitionsDefinitions

ElementElement EventEvent Universe of possibilitiesUniverse of possibilities Venn diagramVenn diagram

Exercise Exercise

In examining wrong side In examining wrong side surgeries in our hospital, what surgeries in our hospital, what are the elements, events and are the elements, events and the universe of possibilities? the universe of possibilities? Draw the Venn DiagramDraw the Venn Diagram

Two Events OccurringTwo Events Occurring

Probability of One or Other Probability of One or Other Event OccurringEvent Occurring

Probability of One or Other Probability of One or Other Event OccurringEvent Occurring

P(A or B) = P(A) + P(B) - P(A & B)

Example: Who Will Join Example: Who Will Join Proposed HMO?Proposed HMO?

P(Frail or Male) = P(Frail) - P(Frail & Male) + P(Male)

ExerciseExercise

3 computers have spies and virus

All computers = 250

Computers with a virus =5

Computers with spies =80

Probability of Two Probability of Two Events co-occurringEvents co-occurring

ExerciseExercise

All computers = 250

Computers with a virus =5

Computers with spies =80

3 computers have spies and viruses

Effect of New KnowledgeEffect of New Knowledge

If A has occurred, the universe of possibilities

shrinks

Conditional ProbabilityConditional Probability

Example: Hospitalization rate of Example: Hospitalization rate of frail elderlyfrail elderly

ExerciseExercise

All computers = 250

Computers with a virus =5

Computers with spies =80

3 computers have spies and viruses

OddsOdds

eventofobabilityeventofobability

eventanofOddsPr1

Pr

eventofOddseventofOdds

eventanofobability

1

Pr

OddsOdds

eventofobabilityeventofobability

eventanofOddsPr1

Pr

90.Pr eventanofobability

90.0190.0

eventanofOdds

OddsOdds

Odds Probability

1:1 0.50

2:1 0.67

3:1 0.75

10:1 0.91

100:1 0.99

Sources of DataSources of Data

Objective frequencyObjective frequency    Subjective opinionsSubjective opinions of experts  of experts 

Forcing Opinions to Behave Forcing Opinions to Behave Like ProbabilitiesLike Probabilities Subjective probabilities can meet Subjective probabilities can meet

axioms of probabilityaxioms of probability Some event must occur Some event must occur

Always trueAlways true Probability must be zero or larger Probability must be zero or larger

Set by conventionSet by convention Probability of complement is one minus Probability of complement is one minus

the probability of the event the probability of the event Forced to meet even when estimates vary.Forced to meet even when estimates vary.

Probability of mutually exclusive events Probability of mutually exclusive events is the sum of probabilities of each event is the sum of probabilities of each event

Choose to meet this assumptionChoose to meet this assumption

Probabilities Provide a Probabilities Provide a Context to Study BeliefsContext to Study Beliefs

Rules of probability provide a Rules of probability provide a systematic and orderly method systematic and orderly method

Two Ways to Assess Subjective Two Ways to Assess Subjective ProbabilitiesProbabilities

Strength of Beliefs Strength of Beliefs Imagined Frequency Imagined Frequency

Uncertainty for rare, one time events can be

measured

An Example of Strength of An Example of Strength of BeliefBelief

On a scale from 0 to 100, where On a scale from 0 to 100, where 100 is for sure, how certain are 100 is for sure, how certain are you that medication errors will you that medication errors will occur in next visit?occur in next visit?

An Exampled of Imagined An Exampled of Imagined FrequencyFrequency

Out of 100 visits, how many Out of 100 visits, how many have had medication errors?have had medication errors?

ExerciseExercise

Ask a question (using strength of belief) Ask a question (using strength of belief) that would assess the probability of wrong that would assess the probability of wrong side surgery in infants in our hospital?side surgery in infants in our hospital?

Ask a question (using imagined Ask a question (using imagined frequencies) that would assess the frequencies) that would assess the probability of wrong side surgery among probability of wrong side surgery among the elderly in our hospital?the elderly in our hospital?

Ask a question that would assess the Ask a question that would assess the probability of medication error in infants or probability of medication error in infants or elderly in our hospital?elderly in our hospital?

Check if the answers meet the axioms of Check if the answers meet the axioms of probability and adjust if they do not.probability and adjust if they do not.

Take Home LessonTake Home Lesson

Probability of events can be measured Probability of events can be measured in subjective or objective waysin subjective or objective ways

What Do You Know?What Do You Know?

Draw a Venn Diagram showing the Draw a Venn Diagram showing the probability of a computer being infected probability of a computer being infected with a virus or a spy.with a virus or a spy.

Estimate the probability of either event and Estimate the probability of either event and both events by interviewing a student.both events by interviewing a student.

What type of question did you ask to What type of question did you ask to assess the probabilities?assess the probabilities?

Calculate the following:Calculate the following: Probability of either, or both event occurring.Probability of either, or both event occurring. Probability of virus infection in computers that Probability of virus infection in computers that

have a spy.have a spy. Probability of virus infection in computers that do Probability of virus infection in computers that do

not have a spy. not have a spy. Odds of neither a spy nor a virus infection. Odds of neither a spy nor a virus infection.