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Bayes

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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
    Business heading
  • Same for no
  • Added a new column

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
    Business heading
  • Same for no
  • Added a new column

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
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