Understanding Basic Probability Concepts and Their Applications

1. Project 1 Lecture Notes

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

3. 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

4. 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

5. 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?

6. 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) =

7. Basic Probability • Properties of Probability • 0≤P(E)≤ 1 for any event E • If E is guaranteed to occur, then P(E)=1 • If E and F cannot happen at the same time, then P(E or F) = P(E) + P(F)

8. 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 (iii) is equivalent to P(S)=1

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

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

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

12. 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 EandF.

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

14. 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.

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

16. Basic Probability • Properties (iv) and EF

17. Basic Probability • The last statement means property (iii) can be rewritten as: • 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

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

19. Basic Probability • DeMorgan’s Laws: F E EC FC

20. Basic Probability • DeMorgan’s Laws: So everything minus the intersection F E EC FC

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

22. 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.

23. 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.

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

25. 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.

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

27. 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.

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

29. 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.

30. 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. HC

31. Basic Probability • Correct & Incorrect notation: Correct Incorrect

32. 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

33. 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

34. Basic Probability • Focus on the Project: • Range: G11:G8236 • Criteria: “yes” • Range: G11:G8236 • Criteria: “no”

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

36. Basic Probability • Focus on the Project:

37. 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