# Probability of Rare Events

In probabilistic risk analysis, the analyst often faces situations where the event of interest is quite rare (less than 5% chance of occurrence); perhaps it has happened only once in a decade.  This review focuses on how to accurately assess probabilities of rare events.

In general there are two ways of assessing probabilities, both of which are not reasonable for assessment of rare probabilities.  The most common approach is to rely on observed frequency of the event.  This method cannot be applied to rare events as by definition rare events do not occur often and one has to accumulate a large data set before reliable estimates can be made.  For an event that occurs once a decade, one has to collect several decades of data before a reliable estimate can be obtained.

Alternatively, many rely on experts to assess probability of events.  But human beings are notoriously ill equipped to distinguish among very small probabilities.  In estimating rare probabilities, sometimes orders of magnitudes are missed; and probability of 1 in million is estimated as 1 in thousand.   An alternative is needed that overcomes difficulties in estimation of rare probabilities.�

# Calibration

Before proceeding, it is important to clarify how would anyone know if the assessed probability of a rare event is accurate.  In general, accuracy of probabilistic forecasts are verified by calibration:  in numerous occasions in which the same probability is forecasted, the frequency of occurrence of the event is compared to the estimated probability.   For example, suppose a weather forecaster predicts that there is 80% chance of rain.  If it rains tomorrow, is this a reasonable forecast?  If it does not rain, is the forecast erroneous?  Neither of these questions can be answered.  In most situations, a single event cannot tell us much about the frequency of that event.  The accuracy of the forecast can only be established if in a large number of days, say 100 days, in which the weather forecaster has predicted 80% chance of rain, it does indeed rain for 80 days.  Only then we can claim that weather forecaster is well calibrated and accurate.

Obviously, the requirement to observe a large number of similar forecast makes it difficult to verify calibration of forecast of rare events: There are not enough such forecasts or observation of the event to compare the two.  So how could one assess the reasonableness of probability estimates for rare events?   In the case of rare events, it may be possible to assess accuracy of the probability of the rare event with a single observation to the contrary.  If a rare event occurs more frequently, we may have to revise our assessment of it.  If an event is expected to occur once every 100,000 occasions, then observing the occurrence of the event after 25 occasions will signal a problem with the estimate.  For example, Cooke (1991) reports that administrators of NASA had predicted the probability of shuttle failure at one in every 100,000 occasions.  Colglazier and Weatherwax (1983) had predicted such failure at one in every 35 flights.  When the Challenger Space Shuttle failed after only 25 flights, that NASA administrators were wrong in assuming shuttle failures would be very rare.

# History

In recent years, there have been many occasions in which risks of rare events have been assessed and subsequent events have helped confirm the accuracy of the risk analysis or improve aspects of the analysis.  Probabilistic risk analysis originated in aerospace industry.  One of the earliest comprehensive studies was started after the loss of life due to a fire in Apollo flight AS-204 in 1967.  In 1969, the Space Shuttle Task Group in the Office of Manned Space Flight of NASA suggested that the probability of loss of life should be less than 1 percent.  Colglazier and Weatherwax (1983) conducted a probabilistic risk analysis of shuttle flights.  But overtime, NASA administrators abandoned numerical forecast of risks as the projected risks were so high to undermine the entire viability of the operations.  Cooke (1991) and Bell and Esch (1989) report that NASA administrators "felt that the numbers could do irreparable harm."  But subsequent shuttle accidents returned the emphasis to probabilistic risk analysis.  Today almost all components of space shuttle go through independent risk analysis (Safaie 1991, 1992, 1994; Hoffman 1998; Planning Research Corporation, 1989, Science Applications International Corporation, 1993, 1995).  A good example of such risk analysis can be found in the work of Pate-Cornell and Fischbeck (1993, 1994), where they assessed the risk of tiles breaking away from the shuttle.  In this award winning study, the authors linked management practices to risks of various tiles on the shuttle.

In nuclear safety, several studies have focused on reactor safety.  The first such study was the Reactor Safety Study (1975).  The study was followed by a series of critical reviews (Environmental Protection Agency, 1976; Union of Concerned Scientists, 1977, American Physical Society, 1975), including in 1997 a Congressional bill to mandate a review panel  to examine the limitations of the study.  The near failure of reactor core at Three Miles Island, however, proved that the scenarios anticipated in the study were indeed correct, though the probability of human failures were underestimated.   Not surprisingly, reviews of Three Miles Island re-emphasized the need for conducting probabilistic risk analysis (Rogovin and Frampton, 1980; Kemeny et al. 1979).  Kaplan and Garrick (1981) conducted a study of probability of reactor melt down.  In 1983, the U.S. Nuclear Regulation Commission issued a manual for how to conduct Probabilistic Risk Analysis.     Probabilistic risk analysis has also been used by the energy firms not focused on nuclear power to predict catastrophic events (Cooke, Jager 1998; Rasmussen, 1981; Ortwin, 1998)

Probabilistic risk analysis has been applied to a variety of natural disasters including earthquake predictions (Chang, Shinozuka, Moore 2000), predicting floods and coastal designs (Voortman, van Gelder, Vrijling 2002; Mai, Zimmermann, 2003; Kaczmarek 2003 ), environmental pollution (Slob, Pieters 1998; Moore, Sample, Suter, Parkhurst, Scott, 1999).  A large number of studies focus on waste disposal and environmental health (Ewing, Palenik, Konikow 2004; Sadiq, Husain, Veitch, Bose. 2003; Cohen 2003; Garrick, Kaplan 1999).   In health care probabilistic risk analysis has focused on analysis of root causes of sentinel adverse events such as wrong side surgery or failure mode and effect analysis of near catastrophic events (Bonnabry, et. al 2005).   Amgen pharmaceutical has also used the procedure for deciding on new product development (Keefeer, 2001).  In failure mode analysis within health care most often the rank order of rare probabilities are assessed and the magnitude of the probability is ignored (DeRosier, Stalhandske, Bagian, Nudell 2002).

The application to terrorism is new.  Taylor, Krings and Alves-Foss (2002) have applied probabilistic risk analysis to assessment of cyber terrorism risks.  Others have suggested the use of these techniques in assessment of terrorism ( Apostolakis, Lemon 2005; Haimes, Longstaff 2002).

# Alternative Approaches to Assessing Probability of Rare Events

There are a number of methods available for assessing probability of rare events.  This review discuses four approaches:  use of fault trees, similarity judgments, importance sampling, and time to the event.  Each of these approaches are further discussed below.

# Fault Trees

The concept of fault trees and reliability trees has a long history in space and nuclear industry.  Several books (Krouwer, 2004) and papers describe this tool (Marx and Slonim, 2003).  The first step in conducting fault trees is to identify the sentinel adverse event that should be analyzed.  Then all possible ways in which the sentinel event may occur is listed.  It is possible that several events must co-occur before the sentinel event may occur.  For example, in assessing the probability of an employee providing information to outsiders, several events must co-occur.  First the employee must be disgruntled.  Second, information must be available to the employee.  Third, outsiders must have contact with the employee.  Fourth, the employee must have a method of transferring the data.  All of these events must co-occur before hospital data is sold to an outside party.  None of these events are sufficient to cause the sentinel event.  In a fault tree, when several events must co-occur, we use an "And" gate to show it.  Each of these events can, in part, depend on other factors.  For example, there may be several ways to transfer the data:  on paper, electronically by email, or electronically on disk.  Any one of these events can lead to transfer of data.  In fault tree when any one of a series of events may be sufficient by themselves to cause the next event to occur, we show this by an "Or" gate.  Fault tree is a collection of events connected to each other by "and" and "Or" gates.  Each event depends on a series of other related events, providing for a complex web of relationships.  A fault tree suggests a robust work process when several events must co-occur before the catastrophic failure occurs.  The more "And" gates are in the tree structure, the more robust the work process modeled.  In contrast, it is also possible for several events by themselves to lead to catastrophic failure.  The more "Or" gates in the path to failure, the less robust the work process.

The second step is to estimate probabilities for the fault tree.  Since the catastrophic failure is rare, it is difficult to asses this probability directly.  Instead, the probability of various events leading to this failure are assessed.  For example, the probability of a finding a disgruntled employee can be assessed.  The probability of an employee having access to large data sets can be assessed by counting employees who have such access during the course of their work.  The probability of an employee being approached by someone to sell data can be assessed by providing an expert data on frequency of reported crimes and asking him/her to estimate the additional unreported rate.  In short, through objective data or subjective opinions of experts various probabilities in the fault tree can be assessed.   The fault tree can then be used to assess the probability of the catastrophic and rare event using the following formula:

Pcatastrophic failure = ij pi,j

In the above formula, "j" represents all events that are related to each other through an "And" gate and "i" represents all events that are related to each other through an "Or" gate.

# Similarity Judgments

Sometimes, we are trying to predict an event that has no precedence but in some way and shape is similar to a previous rare event.  For example, prior to September 11����{� th����{�  attack on skyscrapers in New York city, terrorist tried to attack Eiffel tower by driving a hijacked plane into it.   The two incidences are similar in the sense that both are tall building, which have important symbolic values.  Both were attacked by a passenger jet, hoping that the jet fuel will lead to additional destruction.  They are of course also different incidences occurring for different reasons at different times in different places.  Should an analyst deduce from the attack on Eiffel tower that other similar attacks are likely?

Consider another situation.  Recently, there has been an attack by terrorists on a school, where children were taken hostage and and surrounded by bombs.  Is it possible that a similar attack may occur in a hospital in United States and if so what is the probability of the attack.    The answer to this question depends on two factors.  First, what is the probability of an attack on a school?.  Second, how similar is the hospital scenario to the school situation?

Similarity judgments can be used to extend probability of known rare events to new situations.  Psychologists have conducted numerous experiments showing that similarity of two situations will depend on features they share and features unique to each case (����{� Mobus, 1979; Siegel, McCord, Crawford 1982; Schwarz, Tversky 1980; Catrambone, Beike, Niedenthal 1996).  In 1997,����{� Tversky summarized the research on similarity and provided a mathematical model for judgments of similarity.  According to procedure suggested by Tversky s����{� imilarity of two situations "i" and "j" can be assessed by listing the following three categories of features:

1. Features in the index case but not in the comparison case, fi, not j
2. Features in the comparison case but not in the index case, fnot i, j
3. Features in both cases, fi,j

Then similarity can be measured as the count of shared and not shared features using the following formula:

Sij = fi,j / [fi,j + a fi, not j + b fnot i, j]

In above formula, the constant "a" and "b" add up to 1 and are set based on whether the index case is defining prototype.  These constants if different from .5 allow the comparison case to be more like the index case than vice versa.  For example, they allow a child to be more like father than the father like the child.  For example, consider the similarity between the attack on a hospital in America and attack on the school in Russia.  First, we list features shared or unique across the two situations:

1. Features in the index case (attack on school) and not in the comparison case (attack on hospital)
• No proximity defense, easy access
• No communication system available between rooms allowing terrorist time to collect large number of people
• School age children.
2. Features in the hospital attack scenario and not in the school case.
• Difficulty in gathering the population into one central location
• Availability of security officers
3. Features shared in both
• Focused on vulnerable population
• An ongoing war leading to occupation of the region

While this list is brief, it highlights the procedure.  Once the list has been created the similarity of the two situations can be measured using the formula.  Assume that we let the constant "a" be 0.80 and the constant "b" be 0.20.  Then the similarity of the hospital situation to the school is calculated as:

Similarityschool, hospital = 2 / [2 + 0.80 (2) + 0.20 (3)] = 0.5

Please note that this is not the same as similarity of the school to the hospital, which is:

Similarityschool, hospital = 2 / [2 + 0.80 (3) + 0.20 (2)] = 0.4

Based on this calculation, if we think that the probability of an attack on the school is, lets say 1 in 10,000; then the probability of attack on the hospital is:

Probability of attack on hospital = (1/10000) * 0.5 ≈ 5 in 100,000

# Importance Sampling

One method of improving accuracy of estimates of rare events, is to purposefully examine the event in artificially constructed samples where the event is not rare (Heidelberger 1995, Glynn, Iglehart 1989; Srinivasan, 2002).  Then the frequency of the event in the sample can be extrapolate to the remaining situation proportional to how narrowly the sample was drawn.  The procedure is generally known as importance sampling and involves sampling data from situations where we expect to find the rare event.  Assume that we have taken "M" narrowly defined samples and sample "i" represents Wi cases in the population of interest.  If Pi is the probability of the event in the narrowly defined sample, then probability of the rare event, P,  can be calculated as:

P = (i, …, M Wi Pi)/i=1, …, M Wi

An example may demonstrate this concept.  Suppose we want to estimate the probability of a successful theft of data by overcoming password protection in a computer.  For most organization such an attack is rare, but the attack is more likely to be seen in computers that are infected by a virus.  Suppose in an organization 1 in 100 computers have a major virus.  Also suppose that examination of data trails in these infected computers show that 0.3% involve loss of data.  What is the probability of loss of data anywhere in the organization?  This probability is calculated by weighting the narrow sample of infected computers to reflect the proportion of these computers inside the organization:

P = (1/100) * 0.003 + (99/100) * 0

Note that in this calculation we have assumed that loss of data does not occur in computers without virus infection.  This may be wrong but as a first approximation may be a reasonable step as we have anticipated that most data loss occurs among infected computers.   The importance weighting procedures requires us to know a priori, with high level of certainty, both the conditions under which the rare event are more likely to occur and the prevalence of the conditions.

A special case of importance sampling arises when we over sample rare events so that we can better understand the relationship of causes within these events.  Sometimes extant databases exist collecting information about causes of various sentinel events over the years.  Sometimes, the data is no more than a list of what happened.  To go from these limited data to a joint distribution of causes and sentinel event, we need to remove the effect of over sampling from the data. A revised count of the data is calculated by following these steps:

1. Count the frequency of sentinel event "s" in your databank, call this Cs

2. Count the frequency of causes "i" among the sentinel event "s", call this Cis.

3. Count the number of normal events, Cn,, these are the events in which the sentinel event has not ocurrred.  These data are readily available as they are common everyday occurrences.

4. Estimate the odds for occurrence of sentinel events, shown as Os for sentinel event s.  Typically, the odds is estimated from time to sentinel event procedures described in the following section.  Note that the odds is typically several order of magnitude different from ratio of Cs/Cn

5. Adjust the counts to reflect the expected proportion of sentinel event.  We had over sampled the sentinel events. This adjustment will reduce the count to reflect the proportion we observe the sentinel event.  The revised count of causes among sentinel event, Ris, is provided by the following formula:

An example can describe this procedure.  Suppose we want to examine the joint distribution of wrong blood transfusion and under staffing of the operating room.  Wrong blood transfusion is quite rare, suppose that in the last 1000 operations, it has occurred only 3 times.  In each occasion an investigation was done of various causes.  Two of these occurred when we were understaffed and one when we were not.  Most patients have the right blood transfusion.  In a sample of 500 operations without wrong blood transfusion, 250 were operated on when there was a staff shortage.  With this data we can now estimate the joint distribution of operating room understaffing and wrong blood transfusion.  Table 1 shows the operating room staffing within groups that had or did not have wrong blood transfusion.  Note that these numbers make sense as a column but cannot be added as a row, as the population of patients with wrong blood transfusion have been over-sampled.

 Patients with no wrong blood transfusion in last 500 operations Wrong Blood Transfusion in 1000 operations Total Understaffed Operating Room No 250 1 Not available Yes 250 2 Not available Total 500 3 Not available Table 1:  Count within Types of Blood Transfusion

To create a Table that is not based on separate cohorts, we start with an arbitrary large number of normal operations, say 10000,  who have not had any wrong blood transfusions and distribute these operations based on observed rates of understaffed operating rooms.  The result of this step are shown in Table 2:

 Wrong Blood Transfusion No Yes Total Understaffed Operating Room No 5000 Yes 5000 Total 10,000 Table 2:  Distribute a Large Number of Operations for Normal Patients

Note that the odds for wrong blood transfusion is 3/997.  Using the data and the formulas provided above we can estimate the frequency of wrong blood transfusion cases that can be expected in 10000 operations.  The results are shown in Table 3:

 Wrong Blood Transfusion No Yes Total Understaffed Operating Room No 5000 Yes 5000 Total 10,000 30 Table 3:  Estimate Expected Number of Wrong Blood Transfusions

Next we distribute the estimated number of wrong blood transfusion based on the observed staffing in the cohort of patients with wrong blood transfusion.   The results are shown in Table 4.

 Wrong Blood Transfusion No Yes Total Understaffed Operating Room No 5000 10 Yes 5000 20 Total 10,000 30 Table 4:  Distribute Number of Wrong Blood Transfusions across Operating Room Status

Note that now the row totals can be calculated as the effects of over sampling of wrong blood transfusions has been taken out.  Table 5 shows the joint distribution of wrong blood transfusion and operating room staffing calculated from the original cohort data.

 Wrong Blood Transfusion No Yes Total Understaffed Operating Room No 0.499 0.001 0.500 Yes 0.499 0.002 0.501 Total 0.997 0.003 1.000 Table 5:  Joint Distribution of Blood Transfusion & Operating Room Staffing

From these data we can now estimate the probability of various events including the probability of wrong blood transfusion with and without understaffed operating rooms:

p(Wrong blood transfusion | Understaffed operating room) = 0.004

p(Wrong blood transfusion | No understaffed operating room) = 0.002

Note that there is a 2 fold increase in the probability of wrong blood transfusion if the operating room is understaffed.  Knowing that the operating room is understaffed would increase the likelihood of wrong blood transfusion by odds of 1.33 to 1.

Likelihood ratio associated with understaffed operating room =  p(Understaffed operating room | Wrong blood transfusion) / p(Understaffed operating room | No wrong blood transfusion) = 1.33

Note that we were able to do this analysis from very scant data on an a rare event.

# Time to Event

A method that can allow us to examine rare events directly is through examination of time to the event.  If we assume that an event has a Bernoulli distribution (i.e. the event either happens or does not happen, it has a constant probability of occurrence, and the probability of the event does not depend on prior occurrences of the event); then number of consecutive occurrences of the event has a Geometric distribution.  In a geometric distribution, probability of a rare event, p, can be estimated from the average time to the event, t, using the following formula:

p = 1 / (1+t)

Table 7 shows how this relationship can be explored to calculate rare probabilities.  The expert is asked to provide the dates for the last few times the event has occurred in the last year or decade.  The average time to reoccurrence is calculated and the above formula is used to estimate the probability of the event.

 ISO 17799 word Frequency of event Calculation Rare  probability Negligible Once in a decade =1/(1+3649) 0.0003 Very low 2-3 times every 5 years =2.5/(5*365) 0.0014 Low <= once per year =1/(364+1) 0.0027 Medium <= once per 6 months =1/(6*30+1) 0.0056 High <= once per month =1/(30+1) 0.0333 Very high => once per week =1/(6+1) 0.1429 Extreme => one per day =1/1 1 Table 7:  Estimating probabilities from time between events

For example, suppose we want to know what is the probability of an a terrorist attack in city of Washington DC.�  To calculate this probability, we need only to record the dates of the last attacks in the city and average the time between the attacks.�  This average time between the reoccurrence of the event can then be used to estimate the probability of another attack.

For another example, suppose we do not know the frequency of medication errors in our hospital.�  Furthermore, suppose that last year there were two reports of medication errors, one at start of the year and one in the middle of the year.�  The pattern of medication error suggests 6 months time between errors.�  Average time between errors allows us to estimate the daily probability of medication error:

P( Error) = 1 / (1+6*30) = 0.0056

# Validity of Low Probability High Consequence Estimates

Since there are no practical ways of observing very low probability events, it is difficult to evaluate the accuracy of our estimates.�  Obviously, it is possible, that a contrary event (for example accidents occurring with more frequency than expected) will point out the inaccuracies in our estimation procedure. But in the absence of these contrary events, it is difficult to validate the probabilistic risk analysis findings.�  To improve confidence in the assessment, any or all of the following additional steps can be taken:

• Check the assumptions of the model.

For example, in fault tree often a series of events are linked to each other.�  One could check that this is a reasonable link by examining the conditional independence among these serially linked events.�  In an non-cyclical path if A is shown to affect B and B is shown to affect C, then C should be independent of A given a specific value for B.� �  Conditional independence can be checked by examining partial correlation between A and C, by querying the expert, or by examining causal graphs drawn by experts.

• Check the accuracy of the parameters of the model

While it is not easy to measure the catastrophic event, it is possible to observe the probability of various events used in the model. If these estimates are accurate, we have more confidence in the resulting prediction of the model.

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# Presentations

To assist you in reviewing the material in this lecture, please see the following resources:

1. Listen to lecture on subjective probabilities  (SWF file)

2. Listen to the lecture on independence & Bayes odds form (SWF files)

3. Listen to the lecture on assessing rare probabilities.  Listen to an older version of the same lecture (SWF file)  See the slides for the lecture.  Listen to the same lecture on assessment of rare probabilities broken up in a multi-part series and posted to YouTube:

Part 1:  Introduction and use of fault trees

Part 2:  Assessing Rare Events Using Similarity Judgments

Part 3:  Probability Assessment Using Importance Sampling

Part4:  Probability Assessment Using Time to Event

Narrated lectures require use of Flash.

Copyright © 2006 by Farrokh Alemi, Ph.D.  Created on Tuesday October 4th, 2006.  Most recent revision 10/22/2011. This page is part of a Course on Risk Analysis