Conditional Probability

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Conditional Probability
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Conditional Probability

• So far for the loan project, we know how to
• Compute probabilities for the events in the
  sample space S success, failure.
• Use the loan values to compute the expected value
  of a workout
• Use database filtering to get the information you
  need from the loan records
• What we havent learned yet is how to use the
  characteristics of the borrower education,
  experience, and economic conditions!

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Conditional Probability

• Many of the records are irrelevant for us,
  because they represent borrowers who are very
  different from John Sanders.
• We want to target our computations to the right
  kinds of borrowers.
• This kind of targeting is called conditioning we
  place conditions on the records we consider.

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Conditional Probability

• The basic principle of conditioning is this
• Conditioning permits us to adjust probabilities
  based on new or more specific information, which
  we then take for granted.
• Business can be fast-moving, and new information
  is always coming in we need a way to adapt and
  adjust our expectations based on it.
• Once new information is assimilated, any
  historical data that doesnt fit its pattern may
  be discarded as irrelevant, so our predictions
  can be more accurate to the current situation.

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Conditional Probability

• Think of conditioning as pulling weeds in the
  sample space of a probability experiment.
• When we condition on an event E having happened,
  we eliminate any outcomes outside of E, and
  consider E itself to be the new sample space!

S
E
F
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Conditional Probability

• Notation
• Means the probability of F happening given that E
  has already occurred
• Definition
• In words, this is saying what proportion does F
  represent out of E.

S
E
F
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Conditional Probability

• The formula implies

Notice the reversal of the events E and F
Very Important! These are two different things.
They arent always equal.
Note
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Conditional Probability

• Ex In a classroom of 360 students, 120 students
  play the flute and 120 students are male. There
  are 10 flute-playing males.
• Let E be the event that a randomly-selected
  student is male
• Let F be the event that a randomly-selected
  student plays flute.
• What percentage of male students play the flute?

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Conditional Probability

• Sol The proportion of F that makes up the
  sample space, P(F) . The
  proportion of F that makes up E, however, is P(F
  E) .

S
E
F
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Conditional Probability

• Ex Suppose 22 of Math 115A students plan to
  major in accounting (A) and 67 on Math 115A
  students are male (M). The probability of being
  a male or an accounting major in Math 115A is
  75. Find and .

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Conditional Probability

• Sol
• First find

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Conditional Probability

• Sol

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Conditional Probability

• Sol

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Conditional Probability

• Sometimes one event has no effect on another
• Example flipping a coin twice
• Such events are called independent events
• Definition Two events E and F are independent
  if or

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Conditional Probability

• Implications

So, two events E and F are independent if this is
true.
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Conditional Probability

• The property of independence can be extended to
  more than two events
• assuming that are all
  independent.

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Conditional Probabilities

• INDEPENDENT EVENTS AND MUTUALLY EXCLUSIVE EVENTS
  ARE NOT THE SAME
• Mutually exclusive
• Independence

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Conditional Probability

• Ex Suppose we roll toss a fair coin 4 times.
  Let A be the event that the first toss is heads
  and let B be the event that there are exactly
  three heads. Are events A and B independent?

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Conditional Probability

• Soln
• For A and B to be independent,
• and
• Different, so
• dependent

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Conditional Probability

• Ex Suppose you apply to two graduate schools
  University of Arizona and Stanford University.
  Let A be the event that you are accepted at
  Arizona and S be the event of being accepted at
  Stanford. If and , and your
  acceptance at the schools is independent, find
  the probability of being accepted at either
  school.

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Conditional Probability

• Soln Find .
• Since A and S are independent,

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Conditional Probability

• Soln
• There is a 76 chance of being accepted by a
  graduate school.

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Conditional Probability

• Independence holds for complements as well.
• Ex Using previous example, find the probability
  of being accepted by Arizona and not by Stanford.

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Conditional Probability

• Soln Find .

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Conditional Probability

• Ex Using previous example, find the probability
  of being accepted by exactly one school.
• Sol Find probability of Arizona and not Stanford
  or Stanford and not Arizona.

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Conditional Probability

• Sol (continued)
• Since Arizona and Stanford are mutually
  exclusive (you cant attend both universities)
• (using independence)

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Conditional Probability

• Soln (continued)

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Conditional Probability

• Independence holds across conditional
  probabilities as well.
• If E, F, and G are three events with E and F
  independent, then

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Conditional Probability

• Focus on the Project
• Recall and
• However, this is for a general borrower
• Want to find probability of success for our
  borrower

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Conditional Probability

• Focus on the Project
• Start by finding and
• We can find expected value of a loan work out
  for a borrower with 7 years of experience.

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Conditional Probability

• Focus on the Project
• To find we use the info from the DCOUNT
  function
• This can be approximated by counting the number
  of successful 7 year records divided by total
  number of 7 year records

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Conditional Probability

• Focus on the Project
• Technically, we have the following
• So,

Why technically? Because were assuming that
the loan workouts BR bank made were made for
similar types of borrowers for the other three.
So were extrapolating a probability from one
bank and using it for all the banks.
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Conditional Probability

• Focus on the Project
• Similarly,
• This can be approximated by counting the number
  of failed 7 year records divided by total number
  of 7 year records

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Conditional Probability

• Focus on the Project
• Technically, we have the following
• So,

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Conditional Probability

• Focus on the Project
• Let be the variable giving the value of a
  loan work out for a borrower with 7 years
  experience
• Find

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Conditional Probability

• Focus on the Project
• This indicates that looking at only the years of
  experience, we should foreclose (guaranteed 2.1
  million)

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Conditional Probability

• Focus on the Project
• Of course, we havent accounted for the other
  two factors (education and economy)
• Using similar calculations, find the following

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Conditional Probability

• Focus on the Project

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Conditional Probability

• Focus on the Project
• Let represent value of a loan work out for a
  borrower with a Bachelors Degree
• Let represent value of a loan work out for a
  borrower with a loan during a Normal economy

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Conditional Probability

• Focus on the Project
• Find and

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Conditional Probability

• Focus on the Project
• So, two of the three individual expected values
  indicates a foreclosure

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Conditional Probability

• Focus on the Project
• Cant use these expected values for the final
  decision
• None has all 3 characteristics combined
• for example has all education levels and
  all economic conditions included

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Conditional Probability

• Focus on the Project
• Now perform some calculations to be used later
• We will use the given bank data
• That is is really
• and so on

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Conditional Probability

• Focus on the Project
• We can find
• since Y, T, and C are independent
• Also

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Conditional Probability

• Focus on the Project
• Similarly

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Conditional Probability

• Focus on the Project

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Conditional Probability

• Focus on the Project

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Conditional Probability

• Focus on the Project

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Conditional Probability

• Focus on the Project
• Now that we have found and
  we will use
  these values to find and