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### Bayes Theorem Bayes Theorem An insurance company divides its clients into two categories: those who are accident prone and those who are not. – PowerPoint PPT presentation

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Title: Bayes

1
Bayes Theorem
2
Bayes Theorem
• An insurance company divides its clients into two
categories those who are accident prone and
those who are not. Statistics show there is a
40 chance an accident prone person will have an
accident within 1 year whereas there is a 20
chance non-accident prone people will have an
accident within the first year.
• If 30 of the population is accident prone, what
is the probability that a new policyholder has an
accident within 1 year?

3
Bayes Theorem
• Let A be the event a person is accident prone
• Let F be the event a person has an accident
within 1 year

A
AC
F
4
Bayes Theorem
• Notice weve divided up or partitioned the sample
space along accident prone and non-accident prone

A
AC
F
5
Bayes Theorem
• Notice that and are mutually
exclusive events and that
• Therefore
• We need to find and
• How?

6
Bayes Theorem
• Recall from conditional probability

7
Bayes Theorem
• Thus

P(A) 0.30 since 30 of population is accident
prone
P(FA) 0.40 since if a person is accident
prone, then his chance of having an accident
within 1 year is 40
P(FAC) 0.2 since non-accident prone people
have a 20 chance of having an accident within 1
year
P(AC) 1- P(A) 0.70
8
Bayes Theorem
• Updating our Venn Diagram
• Notice again that

A
AC
F
9
Bayes Theorem
• So the probability of having an accident within 1
year is

10
Bayes Theorem
• Using Tree Diagrams

Accident w/in 1 year P(FA)0.40
Accident Prone P(A) 0.30
No Accident w/in 1 year P(FCA)0.60
Accident w/in 1 year P(FAC)0.20
Not Accident Prone P(AC) 0.70
No Accident w/in 1 year P(FCAC)0.80
11
Bayes Theorem
• Notice you can have an accident within 1 year by
following branch A until F is reached
• The probability that F is reached via branch A is
given by
• In other words, the probability of being accident
prone and having one within 1 year is

12
Bayes Theorem
• You can also have an accident within 1 year by
following branch AC until F is reached
• The probability that F is reached via branch AC
is given by
• In other words, the probability of NOT being
accident prone and having one within 1 year is

13
Bayes Theorem
• What would happen if we had partitioned our
sample space over more events, say , all
them mutually exclusive?
• Venn Diagram

A1
A2
An-1
An
. . . . . . (etc.)
F
14
Bayes Theorem
• For each

15
Bayes Theorem
• Tree Diagram

16
Bayes Theorem
• Notice that F can be reached via branches
• Multiplying across each branch tells us the
probability of the intersection
• Adding up all these products gives

17
Bayes Theorem
• Ex 2 (text tractor example) Suppose there are 3
assembly lines Red, White, and Blue. Chances of
a tractor not starting for each line are 6, 11,
and 8. We know 48 are red and 31 are blue.
The rest are white. What dont start?

18
Bayes Theorem
• Soln.
• R red P(R) 0.48
• W white P(W) 0.21
• B blue P(B) 0.31
• N not starting
• P(N R) 0.06
• P(N W) 0.11
• P(N B) 0.08

19
Bayes Theorem
• Soln.

20
Bayes Theorem
• Main theorem
• Suppose we know . We would like to use this
information to find if possible.
• Discovered by Reverend Thomas Bayes

21
Bayes Theorem
• Main theorem
• Ex. Suppose and partition a space
and A is some event.
• Use
and to determine .

22
Bayes Theorem
• Recall the formulas
• So,

23
Bayes Theorem
• Bayes Theorem

24
Bayes Theorem
• Ex. 4 (text tractor example) 3 assembly lines
Red, White, and Blue. Some tractors dont start
(see Ex. 2). Find prob. of each line producing a
non-starting tractor.
• P(R) 0.48 P(N R) 0.06
• P(W) 0.21 P(N W) 0.11
• P(B) 0.31 P(N B) 0.08

25
Bayes Theorem
• Soln.
• Find P(R N), P(W N), and P(B N)
• P(R) 0.48 P(N R) 0.06
• P(W) 0.21 P(N W) 0.11
• P(B) 0.31 P(N B) 0.08

26
Bayes Theorem
• Soln.

27
Bayes Theorem
• Soln.

28
Bayes Theorem
• Focus on the Project
• We want to find the following probabilities
• and .
• To get these, use Bayes Theorem

29
Bayes Theorem
• Focus on the Project

30
Bayes Theorem
• Focus on the Project
• In Excel, we find the probability to be approx.
0.4774

31
Bayes Theorem
• Focus on the Project
• In Excel,we find the probability to be approx.
0.5226

32
Bayes Theorem
• Focus on the Project
• Let Z be the value of a loan work out for a
borrower with 7 years, Bachelors, Normal

33
Bayes Theorem
• Focus on the Project
• Since foreclosure value is 2,100,000 and on
average we would receive 2,040,000 from a
borrower with John Sanders characteristics, we
should foreclose.

34
Bayes Theorem
• Focus on the Project
• However, there were only 239 records containing
7 years experience.
• Look at range of value 6, 7, and 8 (1 year more
and less)

35
Bayes Theorem
• Focus on the Project
• Use DCOUNT function with an extra Years in
• Same for no

36
Bayes Theorem
• Focus on the Project
• From this you get 349 successful and 323 failed
records
• Let be a borrower with 6, 7, or 8 years
experience
• and

37
Bayes Theorem
• Focus on the Project

38
Bayes Theorem
• Focus on the Project
• Use Bayes Theorem to get new probabilities
• 6, 7, or 8 years, Bachelors, Normal
• (indicates work out)

39
Bayes Theorem
• Focus on the Project
• We can look at a large range of years.
• Look at range of value 5, 6, 7, 8, and 9 (2
years more and less)

40
Bayes Theorem
• Focus on the Project
• Use DCOUNT function with an extra Years in
• Same for no

41
Bayes Theorem
• Focus on the Project
• From this you get 566 successful and 564 failed
records
• Let be a borrower with 5, 6, 7, 8, or 9
years exper.
• and

42
Bayes Theorem
• Focus on the Project

43
Bayes Theorem
• Focus on the Project
• Use Bayes Theorem to get new probabilities
• 5, 6, 7, 8, or 9 years, Bachelors, Normal
• (indicates work out)

44
Bayes Theorem
• Focus on the Project
• Since both indicated a work out while
only indicated a foreclosure, we will work
out a new payment schedule.
• Any further extensions