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
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Bayes Theorem

• For each

15
Bayes Theorem

• Tree Diagram

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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?

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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

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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

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Bayes Theorem

• Focus on the Project

30
Bayes Theorem

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

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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