Probability And Risk - Introduction

1. Probability and risk surround us. Elements of this underlie every decision we make, be it as simple as crossing a road or as major as buying a house. Some basic principles employed in this area will be discussed. The following examples are taken from Paulos (2002). Probability and risk increasingly permeate our lives. Like it or not, we must be able to assess the threats and opportunities that face us. Here's a random sampling of half a dozen hypothetical questions (with answers at the end) inspired by a variety of news stories. Probability And Risk - Introduction Monty Hall 1 Wednesday, 15 October 20143:27 PM

2. Snipers 1. It's impossible to say with any precision what risk the Washington area snipers posed to individuals in suburban Maryland and Virginia, but certainly the likelihood of being attacked was quite small — 13 victims out of about four million people in the affected area over three weeks. Our psychology, however, leads us to be more afraid of what's unfamiliar, out of our control, dramatic, omnipresent, or is the consequence of malevolence. On all these counts, the snipers were more terrifying than more common risks. Probability And Risk - Introduction 2

3. Still, let's consider one of these more common risks. How many traffic fatalities can be expected to occur in any given three-week period in the United States? How many in an area the size of suburban Washington? 2. Early in the sniper case the police arrested a man who owned a white van, a number of rifles, and a manual for snipers. It was thought at the time that there was one sniper and that he owned all these items, so for the purpose of this question let's assume that this turned out to be true. Probability And Risk - Introduction 3

4. Given this and other reasonable assumptions, which is higher — a.) the probability that an innocent man would own all these items or b.) the probability that a man who owned all these items would be innocent? Note your response to 1 and 2. Probability And Risk - Introduction 4

5. Baseball 3. The Anaheim Angels and San Francisco Giants were in this year's World Series. The series ends, of course, when one team wins four games. Is such a series, if played between equally capable opponents, more likely to end in six or seven games? 4. The rules of the series stipulate that team A plays in its home stadium for games 1 and 2 and however many of games 6 and 7 are necessary, whereas team B plays in its home stadium for games 3, 4, and, if necessary, game 5. Probability And Risk - Introduction 5

6. If the teams are evenly matched, which team is likely to play in its home stadium more frequently? Note your response to 3 and 4. Probability And Risk - Introduction 6

7. Elections 5. Eleven million people went to the polls recently in Iraq and, the Iraqi news media assure us, 100 percent of them voted for Saddam Hussein for president. Let's just for a moment take this vote seriously and assume that Hussein was so wildly popular that 99 percent of his countrymen were sure to vote for him and that only 1 percent of the voters were undecided. Let's also assume that these latter people were equally likely to vote for or against him. Given these assumptions, what was the probability of a unanimous 100 percent vote? Probability And Risk - Introduction 7

8. 6. Politics in a democracy is vastly more complicated than it is under dictatorships. Witness the upcoming US elections. What is the probability that the Republicans, the Democrats, or neither will take control of the Senate on Nov. 5? Note your response to 5 and 6. Probability And Risk - Introduction 8

9. Answer to 1. There are approximately 40,000 auto fatalities annually in this country, so in any given three-week period, there would be about 2,300 fatalities. The area around Washington has a population of about four million, or 4/280 of the population of the U.S., so as a first approximation, we could reasonably guess that 4/280 times 2,300, or about 30 auto fatalities, would occur there during any three-week period. Attention must then be paid to the ways in which this area and its accident rate are atypical. Probability And Risk - Introduction 9

10. Answer to 2. The second probability would be vastly higher. To see this, let me make up some illustrative numbers. There are about four million innocent people in the area and, we'll assume, one guilty one. Let's estimate that 10 people (including the guilty one) own all the three of the items mentioned above. The first probability — that an innocent man owns all these items — would be 9/4,000,000 or less than 1 in 400,000. The second probability — that a man owning all three of these items is innocent — would be 9/10. Probability And Risk - Introduction 10

11. Whatever the actual numbers, these probabilities usually differ substantially. Confusing them is dangerous (to defendants). Probability And Risk - Introduction 11

12. Answer to 3. For the World Series to last 6 or 7 games, it must last at least 5 games, at which point one team would be ahead 3 games to 2. If the team that is ahead wins the 6th game, the Series is over in 6 games. If the team that is behind wins the 6th game, the Series goes to 7 games. Since the teams are equally matched, the Series is equally likely to end in 6 or 7 games. Probability And Risk - Introduction 12

13. Answer to 4. The solution requires that we use a bit of probability theory. Doing so, we find that, on average, team A will play 2.9375 games at its home stadium and team B 2.875 games at its home stadium. Thus team A is a bit more likely to play at home. Probability And Risk - Introduction 13

14. Answer to 5. Even given the absurdly generous assumptions above, there would be 110,000 undecided voters (1 percent of 11 million). The probability of a 100 percent vote is thus equal to the probability of flipping a fair coin 110,000 times and having heads come up each and every time! The probability of this is 2 to the power of minus 110,000, or a 1 preceded by more than 30,000 0's and a decimal point. This would be the cosmic mother of all coincidences! Probability And Risk - Introduction 14

15. Answer to 6. As of this writing the Democrats hold a one vote edge in the Senate, and there are a number of races too close to call. Significant consequences will surely flow from small, but unpredictable factors so any prediction won't be ready until Wednesday, Nov. 6. Probability And Risk - Introduction 15

16. To deal with multiple probabilities mathematically we need to refer to Bayes theorem. This type of modelling has become increasingly popular in areas such as modelling for operational risk where there are a large number of variables to consider. Bayes theorem says that if there are n disjoint events (events that have no elements in common) that cover the sample space, then: Probability And Risk - Bayes Theorem 16

17. Where the symbol in the denominator represents the action of taking the sum (or total) of all of the various clauses as set out when added together. The terms are referred to as priors, which are assigned before the samples are chosen, which are associated with a variable. A Venn diagram, to aid in interpreting the theorem, is presented in the figure. Probability And Risk - Bayes Theorem 17

18. Probability And Risk - A Venn diagram for Bayes Theorem To make sense of this, consider the following practical example. 18

19. Two branches of a retail bank, C and D, have each reported eight transactions to risk management for review. Of these, four of the transactions reported by branch C have actually been found to represent errors, whereas six of the transactions reported by branch D were errors. The question, which we shall answer, is given that an error has been identified by senior management as requiring special attention, what is the probability that this originated from branch C? Probability And Risk - A Bayes Theorem Example 19

20. It is known that: Prob (error|C) = 4/8 = 0.5 Prob (error|D) = 6/8 = 0.75 Prob (C) = Prob (D) = ½ (since each branch is equally likely, having each reported eight transactions) Then, using the same mathematical notation as above, Probability And Risk - A Bayes Theorem Example 20

21. Probability And Risk - A Bayes Theorem Example This provides the required result, however we have only so far considered how to combine the results of two branches. The same approach could have been taken for any number of branches, say 70, just by using a greater number of clauses. 21

22. Transactions are reported to risk management from three departments (M1, M2, M3) according to the following table Probability And Risk - Bayes Theorem in Action for More Groups 22

23. If a transaction is found to actually resulting in a loss, what is the probability that department M1 supplied it? From the table the probabilities of actual losses are 0.05, 0.20 and 0.35 respectively. Now using Bayes Theorem, Probability And Risk - Bayes Theorem in Action for More Groups 23

24. It is therefore unlikely that branch M1 will have caused the loss, even though they represent 60% of the population of transactions, since there is a 0.76 probability that the loss was caused by another branch. . Let us consider in a further example the type of analysis that is often undertaken in industries such as the insurance industry. Probability And Risk - Bayes Theorem in Action for More Groups 24

25. In Newcastle 60% of the registered car drivers are at least 30 years old. Of these drivers 4% are annually convicted of a motoring offence. This figure rises to 10% for younger drivers. An employee of your company has recently been charged for speeding. What is the probability the employee is over 30 years old? Probability And Risk - Bayes Theorem Applied to Insurance 25

26. The required probability is Prob(Age30|a motoring offence). To change this into the notation used above we shall let A represent a motoring offence. There are two choices in respect of the variable, the age of the employees. Probability And Risk - Bayes Theorem Applied to Insurance 26

27. These are: These are the priors in this case. It is known that the probability of a motoring offence occurring if the employee is at least 30 is: Prob (A|E1) = 0.04, Probability And Risk - Bayes Theorem Applied to Insurance 27

28. Whereas the probability of a motoring offence occurring if the employee is younger than 30 is: Prob (A|E2) = 0.1, We also know that the probability that the employee is at least 30 and the probability that the employees is younger than 30 are represented by: Prob (E1) = 0.6, Prob (E2) = 0.4 Using Bayes Theorem this gives the following answer: Probability And Risk - Bayes Theorem Applied to Insurance 28

29. It is therefore unlikely that the employee is over 30 years old, which is also why insurance premiums are higher for younger drivers. There is a 37.5% likelihood that they will be at least 30 and a 62.5% likelihood that they are younger than 30. A diagrammatic approach may be used to provide a solution to this type of problem. Probability And Risk - Bayes Theorem Applied to Insurance 29

30. Layman find the theory of simple cumulative risks for example, the risk of food poisoning from the consumption of a series of portions of tainted food, problematic. Problems concerning such risks are extraordinarily difficult for naïve individuals, and the paper by McCloy et al. (2010) explains the reasons for this difficulty. It describes how naïve individuals usually attempt to estimate cumulative risks, and it outlines a computer program that models these methods. Cumulative Risk 30

31. The account predicts that estimates can be improved if problems of cumulative risk are framed so that individuals can focus on the appropriate subset of cases. The authors report two experiments that corroborated this prediction. They also showed that whether problems are stated in terms of frequencies (80 out of 100 people got food poisoning) or in terms of percentages (80% of people got food poisoning) did not reliably affect accuracy. McCloy R., Byrne R.M.J. and Johnson-Laird P.N. 2010 “Understanding Cumulative Risk” The Quarterly Journal of Experimental Psychology 63 499-515. Cumulative Risk 31

32. If a problem involves a sequence of events it is useful to employ a tree diagram to explore the full range of possibilities. This is best demonstrated by the figure, which presents the outcomes for practical example from the table. The information we need to analyse is as follows: Probability And Risk - Tree Diagram 32

33. The circles in the diagram represent events, with the individual points being referred to as nodes. The probabilities associated with the events resulting from any event node must always add to one since each event must actually occur. The individual events are then arranged sequentially, as follows: Probability And Risk - Tree Diagram 33

34. Probability And Risk - A Tree Diagram for the Three Division ErrorReporting Problem 34

35. The branches marked as irrelevant represent the probabilities that a transaction has been reported to risk management but there has not actually been a loss, which are of limited interest to senior management in this context. Looking at the probability of following path M1 then the probability that there is no loss will be 0.6 x 0.95 = 0.57. The six possible probabilities (0.57 + 0.03 + 0.24 + 0.06 + 0.065 + 0.035) must add to one. Probability And Risk - A Tree Diagram for the Three Division ErrorReporting Problem 35

36. However, three of these possible outcomes are irrelevant since the transactions do not result in losses. The remaining terminal probabilities are scaled to achieve a total of one. Therefore the probability that a loss emanates from branch M1 is: Probability And Risk - A Tree Diagram for the Three Division ErrorReporting Problem Exactly as above. 36

37. The Strategy Director has estimated that the probability your company's latest product will be a success is 0.75. • An independent research organisation has been hired to study expected customer reaction to the new product and they also expect the product to succeed. • With hindsight, the research organisation has previously been proved wrong in one survey out of twenty. • Does this new information change the probability of success originally estimated by the Strategy Director? Probability And Risk - Prediction of Success Example 37

38. This example provides the following information to combine: Prob (success predicted) = 0.75 Prob (research correct|success predicted) = 0.95. What you are trying to calculate is the probability that success is predicted given that the research is correct, or: Prob (success predicted|research correct). Probability And Risk - Prediction of Success Example 38

39. This is an exercise again employing Bayes theorem, with the priors being the predictions of success and failure. The first objective is to calculate the following term: Prob (success predicted) x Prob (research correct|success predicted). Using the calculation for independence on the conditional probability Prob (success predicted) x Prob (research correct in success) = 0.75 x 0.95 Probability And Risk - Prediction of Success Example 39

40. for the opposite result the equation becomes: Prob (failure predicted) x Prob (research correct|failure predicted) again using the calculation for independence in the conditional probability Prob (failure predicted) x Prob (research correct in failure) = 0.25 x 0.05 So that: Probability And Risk - Prediction of Success Example 40

41. So that: Probability And Risk - Prediction of Success Example In this particular use of the theorem, where there is independence, a diagram could also be used to derive a solution. 41

42. Probability And Risk - Venn Diagram for the Prediction of Success Example The areas of interest are the two events on the diagonal giving the result obtained above. Alternately tree diagrams could be employed to obtain the required probabilities. 42

43. Probability And Risk - A First Tree Diagram for Prediction of Success 43

44. The original estimation of the Strategy Director is shown first with that of the research agency representing the second branch. Probability And Risk - A First Tree Diagram for Prediction of Success 44

45. It can be applied to each option separately. Probability And Risk - A First Tree Diagram for Prediction of Success 45

46. They could both be correct, both wrong, or one correct and one wrong. Probability And Risk - A First Tree Diagram for Prediction of Success 46

47. Both of these events are independent, so a tree may be drawn with the events in a different sequence. Probability And Risk - A First Tree Diagram for Prediction of Success 47

48. This is particularly useful in deciding which overall events to discard and which to retain. Probability And Risk - A First Tree Diagram for Prediction of Success 48

49. Probability And Risk - A Second Tree Diagram for Prediction of Success 49

50. Here we have reversed the two estimates, putting that of the research agency first and then adding the supporting information from the Strategy Director. The problem is to assess which probabilities to retain from the figures. Clearly “support correct” and “estimate success” should be retained. From the first figure we eliminate “estimate success” and “support false” while from the second figure we eliminate “support correct” and “estimate failure”. Hence the event retained is “support false” and “estimate failure”, a double negative. Probability And Risk - A Second Tree Diagram for Prediction of Success 50