1. Special Type of Conditional Probability
Bayes’ Theorem
Special Type of Conditional Probability

2. Recall- Conditional Probability
P(Y  T  C|S) will be used to calculate
P(S|Y  T  C)
P(Y  T  C|F) will be used to calculate
P(F|Y  T  C)
HOW?????
We will learn in the next lesson?
BAYES THEOREM

3. Definition of Partition
Let the events B1, B2, , Bn be non-empty subsets of a sample space S for an experiment. The Bi’s are a partition of S if the intersection of any two of them is empty, and if their union is S. This may be stated symbolically in the following way.
1. Bi  Bj = , unless i = j.
2. B1  B2    Bn = S.

4. Partition Example S B1 B2 B3

5. Example 1
Your retail business is considering holding a sidewalk sale promotion next Saturday. Past experience indicates that the probability of a successful sale is 60%, if it does not rain. This drops to 30% if it does rain on Saturday. A phone call to the weather bureau finds an estimated probability of 20% for rain. What is the probability that you have a successful sale?

6. Example 1 Events R- rains next Saturday
N -does not rain next Saturday.
A -sale is successful
U- sale is unsuccessful.
Given
P(A|N) = 0.6 and P(A|R) = 0.3.
P(R) = 0.2.
In addition we know R and N are complementary events
P(N)=1-P(R)=0.8
Our goal is to compute P(A).

7. Using Venn diagram –Method1
S=RN
Event A is the
disjoint union of
event R  A
&
event N  A A R N
P(A) = P(R  A) + P(N  A)

8. P(A)- Probability that you have a Successful Sale
We need P(R  A) and P(N  A)
Recall from conditional probability
P(R  A)= P(R )* P(A|R)=0.2*0.3=0.06
Similarly
P(N  A)= P(N )* P(A|N)=0.8*0.6=0.48
Using P(A) = P(R  A) + P(N  A)
= =0.54

9. Let us examine P(A|R) Consider P(A|R)
S=RN R N A
Consider P(A|R)
The conditional probability that sale is successful given that it rains
Using conditional probability formula

10. Tree Diagram-Method 2 Probability Conditional Event Saturday R N
Bayes’, Partitions
Tree Diagram-Method 2
Probability
Conditional
Event
Saturday R N
A R  A 0.3 = 0.06
A N  A 0.6 = 0.48
U R  U 0.7 = 0.14
U N  U 0.4 = 0.32
0.2
0.8
0.7
0.3
0.6
0.4
P(R ) P(A|R)
P(N ) P(A|N)
*Each Branch of the tree represents the intersection of two events
*The four branches represent Mutually Exclusive events

11. Method 2-Tree Diagram
Using P(A) = P(R  A) + P(N  A)
= =0.54

12. Extension of Example1 Consider P(R|A)
The conditional probability that it rains given
that sale is successful
the How do we calculate?
Using conditional probability formula = =
*show slide 7

13. Example 2
In a recent New York Times article, it was reported that light trucks, which include SUV’s, pick-up trucks and minivans, accounted for 40% of all personal vehicles on the road in Assume the rest are cars. Of every 100,000 car accidents, 20 involve a fatality; of every 100,000 light truck accidents, 25 involve a fatality. If a fatal accident is chosen at random, what is the probability the accident involved a light truck?

14. Example 2 Events C- Cars T –Light truck F –Fatal Accident
N- Not a Fatal Accident
Given
P(F|C) = 20/10000 and P(F|T) = 25/100000
P(T) = 0.4
In addition we know C and T are complementary events
P(C)=1-P(T)=0.6
Our goal is to compute the conditional probability of a Light truck accident given that it is fatal P(T|F).

15. Goal P(T|F) Consider P(T|F)
S=CT
Consider P(T|F)
Conditional probability of a Light truck accident given that it is fatal
Using conditional probability formula F C T

16. P(T|F)-Method1 Consider P(T|F)
Conditional probability of a Light truck accident given that it is fatal
How do we calculate?
Using conditional probability formula = =

17. Tree Diagram- Method2 Probability Conditional Event Vehicle C T
F C  F  =
F T  F  =
N C  N 0.6 =
N T N  = .3999
0.6
0.4
0.9998
0.0002

18. Tree Diagram- Method2 = =

19. Partition S B1 B2 B3 A

20. Law of Total Probability
Let the events B1, B2, , Bn partition the finite discrete sample space S for an experiment and let A be an event defined on S.

21. Law of Total Probability

22. Bayes’ Theorem
Suppose that the events B1, B2, B3, , Bn partition the sample space S for some experiment and that A is an event defined on S. For any integer, k, such that
we have

23. Focus on the Project Recall P(Y  T  C|S) will be used to calculate
P(S|Y  T  C)
P(Y  T  C|F) will be used to calculate
P(F|Y  T  C)

24. How can Bayes’ Theorem help us with the decision on whether or not to attempt a loan work out?
Partitions
Event S
Event F
Given
P(Y  T  C|S)
P(Y  T  C|F)
Need
P(S|Y  T  C)
P(F|Y  T  C)

25. Using Bayes Theorem P(S|Y  T  C)  0.477 P(F|Y  T  C)  0.523
LOAN FOCUS EXCEL-BAYES
P(F|Y  T  C)  0.523

26. RECALL
Z is the random variable giving the amount of money, in dollars, that Acadia Bank receives from a future loan work out attempt to borrowers with the same characteristics as Mr. Sanders, in normal times.
E(Z)  $2,040,000.

27. Decision EXPECTED VALUE OF A WORKOUT=E(Z)  $2,040,000
FORECLOSURE VALUE- $2,100,000
RECALL
FORECLOSURE VALUE> EXPECTED VALUE OF A WORKOUT
DECISION
FORECLOSURE

28. Further Investigation I
let Y  be the event that a borrower has 6, 7, or 8 years of experience in the business.
Using the range
Let Z be the random variable giving the amount of money, in dollars, that Acadia Bank receives from a future loan work out attempt to borrowers with Y  and a Bachelor’s Degree, in normal times. When all of the calculations are redone, with Y  replacing Y, we find that P(Y   T  C|S)  and P(Y   T  C|F) 

29. Calculations P(Y   T  C|S)  0.073 P(Y   T  C|F)  0.050
P(S|Y   T  C)  0.558
P(F|Y   T  C)  0.442
The expected value of Z is E(Z )  $2,341,000.
Since this is above the foreclosure value of $2,100,000, a loan work out attempt is indicated.

30. Further Investigation II
Let Y" be the event that a borrower has 5, 6, 7, 8, or 9 years of experience in the business
Let Z" be the random variable giving the amount of money, in dollars, that Acadia Bank receives from a future loan work out attempt to borrowers with 5, 6, 7, 8, or 9 years experience and a Bachelor's Degree, in normal times. Redoing our work yields the follow results.

31. Similarly can calculate E(Z  )
Make at a decision- Foreclose vs. Workout
Data indicates Loan work out

32. Close call for Acadia Bank loan officers
Based upon all of our calculations, we recommend that Acadia Bank enter into a work out arrangement with Mr. Sanders.