What Is Probability?

1. What Is Probability? Farrokh Alemi Ph.D.Professor of Health Administration and PolicyCollege of Health and Human Services, George Mason University4400 University Drive, Fairfax, Virginia 22030703 993 1929 falemi@gmu.edu

2. Lecture Outline • What is probability? • Assessment of rare probabilities • Calculus of probability • Conditional independence • Causal modeling • Case based learning • Validation of risk models • Examples

3. Lecture Outline • What is probability? • Partitioning • Probability axioms • Subjective probability • Hazard functions and related terms • Assessment of rare probabilities • Calculus of probability • Conditional independence • Causal modeling • Case based learning • Validation of risk models • Examples This is a new way of thinking.

4. Why measure uncertainty? • To make tradeoffs among uncertain events  • Measure combined effect of several uncertain events • To communicate about uncertainty

5. Definition • Probability quantifies how uncertain we are about future events

6. More Precise Definition • A probability function assigns numbers to events in a sample space so that: • At least one event from the possible sample must happen. • Probability of any event is greater than or equal to zero. • Probability of a complement of an event is one minus the probability of the event • Probability of two mutually exclusive event occurring is the sum of each

7. What is probability?

8. What is Probability?

9. All non A events What is probability? P( not A ) =

10. Definitions • Element • Event • Universe of possibilities • Venn diagram

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

12. Two Events Occurring

13. Probability of One or Other Event Occurring

14. Probability of One or Other Event Occurring P(A or B) = P(A) + P(B) - P(A & B)

15. Example: Who Will Join Proposed HMO? P(Frail or Male) = P(Frail) - P(Frail & Male) + P(Male)

16. All computers = 250 Computers with spies =80 Computers with a virus =5 3 computers have spies and virus Exercise

17. Probability of Two Events co-occurring

18. Exercise All computers = 250 Computers with spies =80 Computers with a virus =5 3 computers have spies and viruses

19. Effect of New Knowledge If A has occurred, the universe of possibilities shrinks

20. Conditional Probability

21. Example: Hospitalization rate of frail elderly

22. Exercise All computers = 250 Computers with spies =80 Computers with a virus =5 3 computers have spies and viruses

23. Odds

24. Odds

25. Odds

26. Sources of Data • Objective frequency • Subjective opinions of experts 

27. Forcing Opinions to Behave Like Probabilities • Subjective probabilities can meet axioms of probability • Some event must occur • Always true • Probability must be zero or larger • Set by convention • Probability of complement is one minus the probability of the event • Forced to meet even when estimates vary. • Probability of mutually exclusive events is the sum of probabilities of each event • Choose to meet this assumption

28. Probabilities Provide a Context to Study Beliefs Rules of probability provide a systematic and orderly method

29. Two Ways to Assess Subjective Probabilities • Strength of Beliefs • Imagined Frequency Uncertainty for rare, one time events can bemeasured

30. 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?

31. An Exampled of Imagined Frequency • Out of 100 visits, how many have had medication errors?

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

33. Take Home Lesson Probability of events can be measured in subjective or objective ways

34. What Do You Know? • 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. • What type of question did you ask to assess the probabilities? • Calculate the following: • Probability of either, or both event occurring. • Probability of virus infection in computers that have a spy. • Probability of virus infection in computers that do not have a spy. • Odds of neither a spy nor a virus infection.