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 haven’t learned yet is how to use the characteristics of the borrower— education, experience, and economic conditions!

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.

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 doesn’t fit its pattern may be discarded as irrelevant, so our predictions can be more accurate to the current situation.

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! E F S

 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. Conditional Probability E F S

 The formula implies: Notice the reversal of the events E and F Note: Very Important! These are two different things. They aren’t always equal.

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?

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) =. E F S

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.

Conditional Probability  Sol: First find

Conditional Probability  Sol:

Conditional Probability  Sol:

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

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

Conditional Probability  The property of independence can be extended to more than two events: assuming that are all independent.

Conditional Probabilities  INDEPENDENT EVENTS AND MUTUALLY EXCLUSIVE EVENTS ARE NOT THE SAME Mutually exclusive: Independence:

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?

Conditional Probability  Soln: For A and B to be independent, and Different, so dependent

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.

Conditional Probability  Soln: Find. Since A and S are independent,

Conditional Probability  Soln: There is a 76% chance of being accepted by a graduate school.

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

Conditional Probability  Soln: Find.

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.

Conditional Probability  Sol: (continued) Since Arizona and Stanford are mutually exclusive (you can’t attend both universities) (using independence)

Conditional Probability  Soln: (continued)

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

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

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.

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

Conditional Probability  Focus on the Project: Technically, we have the following: So, Why “technically”? Because we’re assuming that the loan workouts BR bank made were made for similar types of borrowers for the other three. So we’re extrapolating a probability from one bank and using it for all the banks.

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

Conditional Probability  Focus on the Project: Technically, we have the following: So,

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

Conditional Probability  Focus on the Project: This indicates that looking at only the years of experience, we should foreclose (guaranteed $2.1 million)

Conditional Probability  Focus on the Project: Of course, we haven’t accounted for the other two factors (education and economy) Using similar calculations, find the following:

Conditional Probability  Focus on the Project:

Conditional Probability  Focus on the Project: Let represent value of a loan work out for a borrower with a Bachelor’s Degree Let represent value of a loan work out for a borrower with a loan during a Normal economy

Conditional Probability  Focus on the Project: Find and

Conditional Probability  Focus on the Project:  So, two of the three individual expected values indicates a foreclosure:

Conditional Probability  Focus on the Project: Can’t 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

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…

Conditional Probability  Focus on the Project: We can find since Y, T, and C are independent Also

Conditional Probability  Focus on the Project: Similarly:

Conditional Probability  Focus on the Project:

Conditional Probability  Focus on the Project:

Conditional Probability  Focus on the Project:

Conditional Probability  Focus on the Project: Now that we have found and we will use these values to find and