What Is Probability? Farrokh Alemi Ph.D. Professor of Health Administration and Policy College of Health and Human Services, George Mason University 4400 University Drive, Fairfax, Virginia
Lecture Outline 1. What is probability? 2. Assessment of rare probabilities 3. Calculus of probability 4. Conditional independence 5. Causal modeling 6. Case based learning 7. Validation of risk models 8. Examples
Lecture Outline 1. What is probability? Partitioning Partitioning Probability axioms Probability axioms Subjective probability Subjective probability Hazard functions and related terms Hazard functions and related terms 2. Assessment of rare probabilities 3. Calculus of probability 4. Conditional independence 5. Causal modeling 6. Case based learning 7. Validation of risk models 8. Examples This is a new way of thinking.
Why measure uncertainty? To make tradeoffs among uncertain events To make tradeoffs among uncertain events Measure combined effect of several uncertain events Measure combined effect of several uncertain events To communicate about uncertainty To communicate about uncertainty
Definition Probability quantifies how uncertain we are about future events Probability quantifies how uncertain we are about future events
More Precise Definition A probability function assigns numbers to events in a sample space so that: A probability function assigns numbers to events in a sample space so that: 1. At least one event from the possible sample must happen. 2. Probability of any event is greater than or equal to zero. 3. Probability of a complement of an event is one minus the probability of the event 4. Probability of two mutually exclusive event occurring is the sum of each
What is probability?
What is Probability?
What is probability? All non A events P( not A ) =
Definitions Element Element Event Event Universe of possibilities Universe of possibilities Venn diagram Venn diagram
Exercise In examining wrong side surgeries in our hospital, what are the elements, events and the universe of possibilities? Draw the Venn Diagram In examining wrong side surgeries in our hospital, what are the elements, events and the universe of possibilities? Draw the Venn Diagram
Two Events Occurring
Probability of One or Other Event Occurring
P(A or B) = P(A) + P(B) - P(A & B)
Example: Who Will Join Proposed HMO? P(Frail or Male) = P(Frail) - P(Frail & Male) + P(Male)
Exercise 3 computers have spies and virus All computers = 250 Computer s with a virus =5 Computers with spies =80
Probability of Two Events co-occurring
Exercise All computers = 250 Computer s with a virus =5 Computers with spies =80 3 computers have spies and viruses
Effect of New Knowledge If A has occurred, the universe of possibilities shrinks
Conditional Probability
Example: Hospitalization rate of frail elderly
Exercise All computers = 250 Computer s with a virus =5 Computers with spies =80 3 computers have spies and viruses
Odds
Odds
Odds OddsProbability 1: : : : :10.99
Sources of Data Objective frequency Objective frequency Subjective opinions of experts Subjective opinions of experts
Forcing Opinions to Behave Like Probabilities Subjective probabilities can meet axioms of probability Subjective probabilities can meet axioms of probability Some event must occur Some event must occur Always true Always true Probability must be zero or larger Probability must be zero or larger Set by convention Set by convention Probability of complement is one minus the probability of the event Probability of complement is one minus the probability of the event Forced to meet even when estimates vary. Forced to meet even when estimates vary. Probability of mutually exclusive events is the sum of probabilities of each event Probability of mutually exclusive events is the sum of probabilities of each event Choose to meet this assumption Choose to meet this assumption
Probabilities Provide a Context to Study Beliefs Rules of probability provide a systematic and orderly method
Two Ways to Assess Subjective Probabilities Strength of Beliefs Strength of Beliefs Imagined Frequency Imagined Frequency Uncertainty for rare, one time events can be measured
An Example of Strength of Belief On a scale from 0 to 100, where 100 is for sure, how certain are you that medication errors will occur in next visit? On a scale from 0 to 100, where 100 is for sure, how certain are you that medication errors will occur in next visit?
An Exampled of Imagined Frequency Out of 100 visits, how many have had medication errors? Out of 100 visits, how many have had medication errors?
Exercise Ask a question (using strength of belief) that would assess the probability of wrong side surgery in infants in our hospital? Ask a question (using strength of belief) that would assess the probability of wrong side surgery in infants in our hospital? Ask a question (using imagined frequencies) that would assess the probability of wrong side surgery among the elderly in our hospital? Ask a question (using imagined frequencies) that would assess the probability of wrong side surgery among the elderly in our hospital? Ask a question that would assess the probability of medication error in infants or elderly in our hospital? Ask a question that would assess the probability of medication error in infants or elderly in our hospital? Check if the answers meet the axioms of probability and adjust if they do not. Check if the answers meet the axioms of probability and adjust if they do not.
Take Home Lesson Probability of events can be measured in subjective or objective ways
What Do You Know? Draw a Venn Diagram showing the probability of a computer being infected with a virus or a spy. Draw a Venn Diagram showing the probability of a computer being infected with a virus or a spy. Estimate the probability of either event and both events by interviewing a student. Estimate the probability of either event and both events by interviewing a student. What type of question did you ask to assess the probabilities? What type of question did you ask to 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 have a spy. Probability of virus infection in computers that have a spy. Probability of virus infection in computers that do not have a spy. Probability of virus infection in computers that do not have a spy. Odds of neither a spy nor a virus infection. Odds of neither a spy nor a virus infection.