Project 1 Lecture Notes

Table of Contents Basic Probability Word Processing Mathematics Summation Notation Expected Value Database Functions and Filtering Conditional Probability Bayes’ Theorem

Basic Probability  Sometimes outcomes are determined by chance  A collection of outcomes is called an event  The probability of an event, denoted P(E), is the likelihood an event E will occur

Basic Probability  P(E) is always between 0 and 1 This means there is between a 0% chance and 100% chance an event E will occur

Basic Probability  Three ways to determine probability Empirically (through trials)  Flip a coin a 100 times. How many times do you expect to see heads? What about a 1000 flips? By Authority (an expert)  Meteorologist says there’s a 30% chance of rain Common Agreement (universally accepted)  Roll a dice. What are your chances of getting a six?

Basic Probability  Empirically-based probabilities mean: The fraction of times an event E occurs in a large number of trials will be very close to P(E)  Universally-based probabilities mean: P(E) =

Basic Probability  Properties of Probability (i)0≤P(E)≤ 1 for any event E (ii)If E is guaranteed to occur, then P(E)=1

Basic Probability  Properties of Probability (cont) The collection of all possible outcomes in an experiment is called the sample space and is denoted by the letter S. So property (ii) is equivalent to P(S)=1

Basic Probability  Venn diagrams: E F The union of E andF, represented by E U F is the collection of items that appear in E or F or in both E and F.

Basic Probability  Venn Diagrams An example:  Let S = {letters in alphabet}  Let V = {vowels}  Let C = {consonants}  Let F = {1 st three letters in alphabet}

Basic Probability The set of vowels The set of 1 st three letters  V U F = { a, b, c, e, i, o, u }

Basic Probability  Venn diagrams: E F The intersection of E and F, represented by E ∩ F is the collection of terms that appear in both E and F.

Basic Probability The set of vowels The set of 1 st three letters  V ∩ F = { a } V ∩ F

Basic Probability  More Properties: The empty set, represented by { }, is the set containing no items. If E ∩ F = { }, then there are no members that appear in both E and F. We say that E and F are mutually exclusive events. They cannot happen both at the same time.

Basic Probability  V ∩ C = { } The set of vowels The set of 1 st three letters

Basic Probability  Suppose we’re rolling a 6-sided die. Let E be the event we roll a 3 AND F be the event we roll an even number.  What’s the sample space, S?  S = { }

Basic Probability  S = { }  E = {3} while F = {2, 4, 6}  What is the probability of rolling either a 3 or an even number?  P(E or F) = P(E) + P(F) = 1/6 + 3/6 = 4/6

Basic Probability  BUT we cannot always do this adding of probabilities.  Why?  Suppose we’re drawing one card from a standard deck. Let E be the event of drawing a face card Let F be the event of drawing a heart.

Basic Probability  Sample Space: A♠A♠ K♠K♠ Q♠Q♠ J♠J♠ 2♠2♠ 3♠3♠ 4♠4♠ 5♠5♠ 6♠6♠ 7♠7♠ 8♠8♠ 9♠9♠ 10 ♠ A♣A♣ K♣K♣ Q♣Q♣ J♣J♣ 2♣2♣ 3♣3♣ 4♣4♣ 5♣5♣ 6♣6♣ 7♣7♣ 8♣8♣ 9♣9♣ 10 ♣ A♥A♥ K♥K♥ Q♥Q♥ J♥J♥ 2♥2♥ 3♥3♥ 4♥4♥ 5♥5♥ 6♥6♥ 7♥7♥ 8♥8♥ 9♥9♥ 10 ♥ A♦A♦ K♦K♦ Q♦Q♦ J♦J♦ 2♦2♦ 3♦3♦ 4♦4♦ 5♦5♦ 6♦6♦ 7♦7♦ 8♦8♦ 9♦9♦ 10 ♦

Basic Probability  Try what we did before:  Now, let’s actually count E or F in sample space

Basic Probability  Sample Space A♠A♠ K♠K♠ Q♠Q♠ J♠J♠ 2♠2♠ 3♠3♠ 4♠4♠ 5♠5♠ 6♠6♠ 7♠7♠ 8♠8♠ 9♠9♠ 10 ♠ A♣A♣ K♣K♣ Q♣Q♣ J♣J♣ 2♣2♣ 3♣3♣ 4♣4♣ 5♣5♣ 6♣6♣ 7♣7♣ 8♣8♣ 9♣9♣ 10 ♣ A♥A♥ K♥K♥ Q♥Q♥ J♥J♥ 2♥2♥ 3♥3♥ 4♥4♥ 5♥5♥ 6♥6♥ 7♥7♥ 8♥8♥ 9♥9♥ 10 ♥ A♦A♦ K♦K♦ Q♦Q♦ J♦J♦ 2♦2♦ 3♦3♦ 4♦4♦ 5♦5♦ 6♦6♦ 7♦7♦ 8♦8♦ 9♦9♦ 10 ♦

Basic Probability  So what happened?  Our counting shows there are 22 cards which can be either a face card or a heart.  We double-counted the overlap!

Basic Probability  In our die example: E: event rolled a 3 F: event rolled an even number S = { }  E and F were mutually exclusive events (both could not happen)

Basic Probability  In our card example: E event you drew a face card F event you drew a heart  E and F were NOT mutually exclusive (both could happen)  So moral of the story is...

Basic Probability  Properties (iv) and E F

Basic Probability  If E and F are mutually exclusive, then P(EUF) = P(E) + P(F)  If E, F, G are pair-wise mutually exclusive, then P(E U F U G) = P(E) + P(F) + P(G) For more events, the process is similar

Basic Probability  More Properties: The complement of an event E, written as E C, is the set of items NOT contained in E. Notice in the last Venn Diagram, C = V C P(E C ) = 1 – P(E)

Basic Probability  DeMorgan’s Laws: E F ECEC FCFC

Basic Probability  DeMorgan’s Laws: E ECEC FCFC So everything minus the intersection F

Basic Probability  DeMorgan’s Laws:  This leads to two more properties: (vi) (vii)

Basic Probability  Ex. Suppose we toss a fair coin 3 times. The sample space is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. What is the probability of getting exactly 2 tails?  Soln. We count all of the times when there are exactly 2 tails: HTT, THT, TTH. Since there are 8 possible outcomes, the answer is 3/8.

Basic Probability  Ex. Suppose the probability of owning a house (H) is 47% while the probability of owning a car (C) is 73%. If the probability of owning a house and a car is 28%, find the probability of owning a house or a car.

Basic Probability  Soln. Therefore, the probability of owning a house or a car is 92%.

Basic Probability  Ex. Suppose the probability of owning a house (H) is 47% while the probability of owning a car (C) is 73%. If the probability of owning a house and a car is 28%, find the probability of not owning a house.

Basic Probability Soln. Therefore, the probability of not owning a house is 53%.

Basic Probability  Ex. Suppose the probability of owning a house (H) is 47% while the probability of owning a car (C) is 73%. If the probability of owning a house and a car is 28%, find the probability of neither owning a house nor owning a car.

Basic Probability  Soln. We want to find, that is no house and no car.

Basic Probability  Ex. Suppose the probability of owning a house (H) is 47% while the probability of owning a car (C) is 73%. If the probability of owning a house and a car is 28%, find the probability of not owning a house and owning a car.

Basic Probability  Soln. We want to find, that is no house and a car. When you want to find “not A intersect B,” draw a Venn diagram. H C

Basic Probability  Correct & Incorrect notation: CorrectIncorrect

Basic Probability  Focus on the Project: Define variables:  S: successful loan work out  F: failed loan work out  Use Loan Records.xls and COUNTIF function in Excel

Basic Probability  Focus on the Project: Range is the collection of cells from which you want to count Criteria is the information you want to count

Basic Probability  Focus on the Project: Range: G11:G8236 Criteria: “yes” Range: G11:G8236 Criteria: “no”

Basic Probability  Focus on the Project: 3818 successful work out situations 4408 failed work out situations 8226 total records

Basic Probability  Focus on the Project:

Basic Probability  Focus on the Project: These probabilities are generally true for the typical borrower However, they do not account for the specific characteristics of John Sanders