Introduction to Probability, Conditional Probability, Bayes's Rule, Example of Sampling Problem using Bayes's Theorem, story from Howard Raiffa

2. A = person entering our clinic
has lung cancer = 10% = 0.1
B = person entering our clinic
is a smoker = 50% = 0.5
Likelihood data = P(B|A) = 0.8
P(A|B) =
(0.8) x (0.1)
(0.5)
= 0.16

3. •
“Bayesian Inference” refers to use of Bayes’
Theorem to update probabilities as new
evidence is available
•
Especially valuable to analyze trends when
data will continue to flow in over time
•
Provides a rational method for updating
beliefs in science, engineering, medicine, law

4. Sets
Venn Diagrams
Terminology
Symbols
Experiment Space S

5. A

6. B

7. A
B

8. Intersection of sets = creatures satisfying
both constraints: two-legged And flying
A
∩
B
the “and” function

9. 
Caution: Venn Diagrams are
Logical relationships,
not the comparative size of sets.
A
B

10. A
B

11. A
B

12. Union: e.g. creatures satisfying either
constraint: two-legged or flying
A

B
the “or” function

13. What if no
intersection?
A
A∩B=
(null set, empty set,
or impossible event)
B

14. S is the set of all possible outcomes
S is often called “the sure event”
EVENT = subset of S
A S
if A is an event, Anot is the event
that A does not occur

15. A  Anot = 
null, impossible set
A  Anot = S
set of all outcomes

16. the probability of A, given B
P(A│B)

21. Ask some lawyers who
try cases on probabilistic
evidence?
Ask political pundits
who make election
predictions?
Ask a class of MBA students
at Harvard B-School
taking a course in
Decision Theory?