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.
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.
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.
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:
P catastrophic failure = ∑i ∏j 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.
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 11th 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 similarity of two situations "i" and "j" can be assessed by listing the following three categories of features:
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:
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:
Similarity school, 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:
Similarity school, 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
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=1, …, 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 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 1 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.
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 Risk Analysis
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:
American Physical Society, Study group on light water reactor safety: Report tot he American Physical Society, Review of Modern Physicians Vol. 47, Supplemental No. 1, 1975.
Apostolakis GE, Lemon DM. A Screening Methodology for the Identification and Ranking of Infrastructure Vulnerabilities Due to Terrorism. Risk Analysis 2005, 25:2, 361-376
Bell TE, Esch K. The space shuttle: A case study of subjective engineering. IEEE Spectrum, 1989, 42-46.
Bonnabry P, Cingria L, Sadeghipour F, Ing H, Fonzo-Christe C, Pfister RE. Use of a systematic risk analysis method to improve safety in the production of paediatric parenteral nutrition solutions. Qual Saf Health Care. 2005 Apr;14(2):93-8.
Catrambone R., Beike D., Niedenthal P. (1996) Is the self-concept a habitual referent in judgments of similarity? Psychological Science; 7 (3): 158-163.
Chang SE, Shinozuka M, Moore JE. Probabilistic Earthquake Scenarios: Extending Risk Analysis Methodologies to Spatially Distributed Systems. Earthquake Spectra, 2000, 16: 3, pp. 557-572
Cohen BL. Probabilistic risk analysis for a high-level radioactive waste repository. Risk Anal. 2003 Oct;23(5):909-15.
Colglazier EW, Weatherwax RK. Failure estimates for the space shuttle. Abstracts for Society for Risk Analysis Annual Meeting 1986, Boston MA, p 80, Nov 9-12, 1986.
Cooke R, Jager E. A probabilistic model for the failure frequency of underground gas pipelines. Risk Anal. 1998 Aug;18(4):511-27.
Cooke RM. Experts in uncertainty: Opinion and subjective probability in science, Oxford university Press, New York, 1991.
DeRosier J, Stalhandske E, Bagian JP, Nudell T. Using health care Failure Mode and Effect Analysis: the VA National Center for Patient Safety's prospective risk analysis system. Jt Comm J Qual Improv. 2002 May;28(5):248-67, 209.
Environmental Protection Agency, Reactor Safety Study Oversight Hearings Before the Subcommittee on Energy and the Environment of the Committee on Interior and Insular Affairs, House of Representatives, 94th Congress, Second Session, Serial No. 84-61, Washington DC, June 11, 1976.
Ewing RC, Palenik CS, Konikow LF. Comment on "Probabilistic risk analysis for a high-level radioactive waste repository" by B. L. Cohen in Risk Analysis, volume 23, 909-915. Risk Anal. 2004 Dec;24(6):1417-1419.
Fox EP. SSME Alternate Turbopump Development Program—Probabilistic Failure Methodology Interim Report. FR-20904-02, 1990.
Garrick BJ, Kaplan S. A decision theory perspective on the disposal of high-level radioactive waste. Risk Anal. 1999 Oct;19(5):903-13.
Glynn PW, Iglehart DL. Importance sampling for stochastic simulations. Management Science 35: 11 (November 1989), 1367 - 1392.
Haimes YY, Longstaff T. The Role of Risk Analysis in the Protection of Critical Infrastructures Against Terrorism. Risk Analysis, 2002, 22:3, pp. 439-444.
Heidelberger P. Fast simulation of rare events in queueing and reliability models. ACM Transactions on Modeling and Computer Simulation (TOMACS) archive 5: 1 43 - 85, 1995
Hoffman CR, Pugh R, Safie FM. Methods and Techniques for Risk Prediction of Space Shuttle Upgrades. AIAA, 1998
Kaczmarek Z. The impact of climate variability on flood risk in Poland. Risk Anal. 2003 Jun;23(3):559-66.
Kaplan S, Garrick B. On the quantitative definition of risk. Risk Analysis, 1981, 1: page 11-27.
Keefeer DL. Practice abstract. Interfaces 31: 5, 2001, pp 62-64.
Kemeny J. Report of the President's Commission on the Accident at Three Mile Island, Washington DC, 1979.
Krouwer JS. Managing Risk In Hospitals Using Integrated Fault Trees And Failure Mode Effects And Criticality Analysis. AACC Press, 2004.
Marx DA, Slonim AD. Assessing patient safety risk before the injury occurs: an introduction to sociotechnical probabilistic risk modelling in health care. Qual Saf Health Care. 2003 Dec;12 Suppl 2:ii33-8.
Mai S, Zimmermann C. Risk Analysis-Tool for Integrated Coastal Planning. Proc. of the 6th Int. Conf. on Coastal and Port Engineering, 2003.
Mobus C. (1979) The analysis of non-symmetric similarity judgments: Drift model, comparison hypothesis, Tversky's contrast model and his focus hypothesis. Archiv Fur Psychologie; 131 (2): 105-136.
Moore, DRJ, Sample BE, Suter GW, Parkhurst BR, Scott TR. A Probabilistic risk assessment of the effects of Methylmercury and PCBs on mink and Kingfishers along East Fork Poplar Creek, Oak Ridge, Tennessee, USA. Environmental Toxicology and Chemistry, 18: 12, pp. 2941-2953, 1999.
Ortwin R. Three decades of risk research: accomplishments and new challenges. Journal of Risk Research, 1998, 1:1 pp 49 - 71.
Pate-Cornell ME, Fischbeck PS. Probabilistic Risk Analysis and Risk--Based Priority Scale for the Tiles of the Space Shuttle. Reliability Engineering and System Safety. Vol. 40, no. 3, pp. 221-238. 1993.
Pate-Cornell ME, Fischbeck PS. Risk management for tiles of the space shuttle. Interfaces, 1994, 24: 1, pp 64-86.
Planning Research Corporation, Independent Assessment of Shuttle Accident Scenario Probabilities for Galileo Mission and Comparison with NSTS Program Assessment, 1989.
Rogovin M, Frampton GT. Three Mile Island, a Report to the Commissioners and to the Public, Government Printing Office, 1980.
Rasmussen NC. The Application of Probabilistic Risk Assessment Techniques to Energy Technologies. Annual Review of Energy, 6: 123-138, 1981.
Sadiq R, Husain T, Veitch B, Bose N. Distribution of arsenic and copper in sediment pore water: an ecological risk assessment case study for offshore drilling waste discharges. Risk Anal. 2003 Dec;23(6):1309-21.
Safie FM. A Statistical Approach for Risk Management of Space Shuttle Main Engine Components. Probabilistic Safety Assessment and Management, 1991
Safie FM. A Risk Assessment Methodology for the Space Shuttle External Tank Welds. Reliability and Maintainability Symposium, 1994
Safie FM, Fox EP. A Probabilistic Design Analysis Approach for Launch Systems. AIAA/SAE/ASME 27th Joint Propulsion Conference, 1991
Safie FM. Use of Probabilistic Design Methods for NASA Applications. ASME Symposium on Reliability Technology, 1992.
Siegel P.S., McCord D. M., Crawford A. R. (1982) An experimental note on Tversky's features of similarity. Bulletin of Psychonomic Society; 19 (3): 141-142.
Schwarz G, Tversky A. (1980) On the reciprocity of proximity relations. Journal of Mathematical Psychology; 22 (3): 157-175.
Science Applications International Corporation, Probabilistic Risk Assessment of the Space Shuttle Phase 1: Space Shuttle Catastrophic Failure Frequency Final Report, 1993
Science Applications International Corporation, Probabilistic Risk Assessment of the Space Shuttle, 1995
Slob W, Pieters MN. A probabilistic approach for deriving acceptable human intake limits and human health risks from toxicological studies: general framework. Risk Anal. 1998 Dec;18(6):787-98.
Srinivasan R. Importance Sampling. Springer, 2002.
Taylor C, Krings A, Alves-Foss J. Risk Analysis and Probabilistic Survivability Assessment (RAPSA): An Assessment Approach for Power Substation Hardening. Proc. ACM Workshop on Scientific Aspects of Cyber Terrorism, 2002.
Tversky A. (1977) Features of similarity. Psychological Review; 84 (4): 327-352.
Union of Concerned Scientists. The risk of nuclear power reactors: a review of the NRC reactor study, WASH-1400, 1977.
U.S. NRC, Reactor Safety study. U.S. Nucler Regulatory Commission, WASH-1400, NUREG-751014, 1975.
U.S. NRC, PRA Procedures Guide, U.S. Nuclear Regulatory Commission, NUREG/CR-2300, 1983.
Voortman HG, van Gelder P, Vrijling JK Risk-based design of large-scale flood defense systems. 28th International Conference on Coastal Engineering, 2002.
Advanced learners like you, often need different ways of understanding a topic. Reading is just one way of understanding. Another way is through writing. When you write you not only recall what you have written but also may need to make inferences about what you have read. Please complete the following assessment:
To assist you in reviewing the material in this lecture, please see the following resources:
Narrated lectures require use of Flash.
Recently Asked Questions
In this section, you will find answers to questions asked by you or others.
Question: When asking two experts to evaluate scenarios, how much detail must be given in each scenario? Could you elaborate, and give me other examples? Answer: Scenarios should be realistic, and should include at least attributes that are used by your model. For example, if a decision is to select/rank purchases, the expert should be provided with what is normally available in such a case.
Question: how can you measure if an event is rare or not? Answer: Most people will consider a probability of less than 5% as indication of a rare event
Suggestions for Improvement
You can suggests changes or see below suggestions made by others:
Comment: Very Good Site http://www.pesolamedia.com/argumentative-essay-on-gun-control/ custom powerpoint presentation The Waterloo, Ontario-based company's steep revenue decline- and mounting losses have revived fears that BlackBerry, once ahigh-flyer and pioneer in the smartphone sector, now faces anignominious death.
Comment: perfect design thanks http://www.qverlondres.com/buy-college-paper/ custom homework on books "I made it a policy since the very first season not to know too far in advance," Cranston said. "So I'll read a script of the next episode about five, six days before we start shooting that episode and that's as close as I get to it. That's when I discover what Walter White is going to do."
Comment: International directory enquiries <a href=" http://www.there-fore.com/index.php/mla-essay-writing ">make an essay</a> During the Goodwood Revival sale on Saturday September 14, another world record was made, this time for a standard road-going E-yype Jaguar – a 1961 Series 1 "flat floor" roadster going for Â£225,500. It had been owned by one family since 1963.
Comment: I work with computers <a href=" http://www.fdt.ie/index.php/sample-termpaper-proposal-for-humanity/ ">essay writing services for free</a> She accepted a job in the women and gender studies department at Hunter. She started a blog, theuniversalcondition.com, which documents her healing process through writing. And she is planning to return to the Caribbean with OÃ¢Â€Â™Connor, whom she is still dating Ã¢Â€Â” this time, to the Bahamas.
Comment: I've just graduated <a href=" http://www.pesolamedia.com/help-writing-a-book/ ">help writing a book</a> Zimmerman shot and killed Martin in Sanford, Fla., on Feb. 26, 2012. Zimmerman, 29, said he shot Martin, 17, in self-defense, while prosecutors argued that Zimmerman "profiled" Martin and concluded he was a criminal.
Comment: Excellent work, Nice Design http://www.qverlondres.com/phd-thesis-writing/ custom writing company 5. Connor Cook for Heisman. I jest. But the Michigan State sophomore continues to impress with a second consecutive strong performance, hitting 22-of-31 passes for 235 yards with two touchdowns and a pick in a 42-28 win vs. Indiana.
Comment: I stay at home and look after the children <a href=" http://www.fdt.ie/index.php/essay-research-paper/ ">write my aper</a> The Mets and Wheeler categorized it as "regular," stiffness that was expected after the 23-year old pushed himself 20 innings past last season's total. There were no doctors involved in the decision, Wheeler has just received normal treatment for the stiffness and was able to play catch Friday.
Comment: I'm a partner in http://www.hungarianbiotech.org/index.php/zithromax-joint-pain order zithromax z pak Ã¢Â€ÂœWe want to work alongside doctors to bring about improvements, but we must all work together to protect the NHS from costly abuse. We want a system that is fair for the British taxpayer by ensuring that foreign nationals pay for their NHS treatment,Ã¢Â€Â a spokesperson said.
Comment: Will I be paid weekly or monthly? <a href=" http://garrygillard.net/blog/?university-assignment-help-uk ">well lever reviews for essay writing services interruption rule</a> Jack was due to appear at the Black Hat hacking conventionin Las Vegas next week, demonstrating techniques for remotelyattacking implanted heart devices. He said he could kill a manfrom 30 feet (9 metres) away.
Comment: How would you like the money? <a href=" http://garrygillard.net/blog/?algebra-homework-help-online-free#funny ">best essay helper</a> The shocking scandals of the mid-1870s – from the Whiskey Ring's tax evasion and bribery of administrative officials to secretary of War William Belknap and his wife's patronage kickbacks – consumed Grant's second term. The investigations reached as high as the president's personal secretary (Orville Babcock) and prevented Grant from focusing on the two major issues severely undermining the state of the newly reunited union: a weak economy and worsening race relations in the South.
Comment: A few months http://www.contravision.de/en/200-mg-topamax-for-migraines.html topamax 200 mg tablet Ã¢Â€ÂœEvery day I have to live with this, no one knows the pain I carry,Ã¢Â€Â she continued. Ã¢Â€ÂœEspecially because it was a misunderstanding, there were so many people in the middle that caused so much pain, so many problems that it was impossible not to be upset.Ã¢Â€Â
Comment: Have you got any qualifications? <a href=" https://badgesforvets.org/mirtazapine-cost-uk.html ">mirtazapine street price</a> Following a mistake in United’s 1-1 draw with Tottenham in January, when a weak punch led to the conceding of an injury-time equaliser, De Gea’s Old Trafford appeared to be on the line with Ferguson considering a summer move for a more experienced goalkeeper.
Comment: One moment, please <a href=" http://www.hungarianbiotech.org/index.php/lotrel-10 ">cheap amlodipine</a> “So what happens after he [the Prince] climbs up and rescues her?” asks Richard Gere to Julia Roberts in Pretty Woman. “She rescues him right back,” comes her beaming reply, before the happy couple descend the fire escape on which their romance is sealed and presumably speed off to spend the rest of their days at a beachfront property in Malibu.
Comment: this is be cool 8) <a href=" http://www.common-sense.at/en/medikament-remeron-soltab-15-mg#becomes ">remeron 15 mg for sleep</a> She also finds that she's more patient with rude clients. When a client was brash with her about a sophisticated trade he wanted done on an extremely tight deadline, Lockwood remained patient and assumed the man's out-of-character behavior was probably due to something in his personal life. That helped her complete the trade to the man's satisfaction, and he remains an important source of her revenue.
Comment: Where do you study? <a href=" http://www.hungarianbiotech.org/index.php/order-keflex#descent ">keflex oral</a> "She was sent home with medication that would reduce her blood pressure and was advised to improve her diet so as not to raise her cholesterol levels and thus decrease the chance of her having a second bleeding episode. She was sedated because the headaches were too sharp," he told Reuters. "We didn't send her home to be sedated and wait until she died in her sleep."
Comment: A company car <a href=" http://www.groenservicebvba.be/indocin-75 ">indocin er 75mg</a> Microsoft's foray into mobile software, however, has been disappointing and the poor performance is under a spotlight amid sluggish demand for personal computers, the lifeblood the Redmond, Wash., company's revenue.
Comment: Where are you calling from? <a href=" http://armisteadguns.com/custom-writing-services-reviews/ ">purchase a literature review</a> "I was just so surprised when I came across the line. When I started the lap, it looked on the screen that I was about seventh or eighth and I was like, 'Oh my god', especially as it was raining more," said the Mercedes driver.
Comment: I'm sorry, she's <a href=" http://paultierney.com/social-work-essays-for-cheap/#gap ">online assignment expert</a> She also described how one flight attendant put a terrified young boy on her back and slid down a slide. Lee put out fires and helped passengers to safety not discovering until much later that she had a broken tailbone, she told the AP.
Comment: An envelope <a href=" http://www.afsbt.org/index.php/proscar-5-mg-hair-loss-women ">2.5mg proscar</a> A dozen survivors remained hospitalized Wednesday, half of them flight attendants, including three thrown from the jet. Meanwhile, other survivors and their family members visited the crash site, where some shed tears and others stood in disbelief, passenger Ben Levy said. They were kept about 50 yards away from the wreckage, which was surrounded by metal railing.
Comment: Could I borrow your phone, please? <a href=" http://elinorlipman.com/generic-ambien-cost-costco.html ">ambien 10 mg vs ambien cr 12.5</a> Plastic baggies are a messÃ¢Â€Â”those plastic flaps practically invite your ham and cheese to explode all over the inside of your bag. This compact box closes tight, has room for a sandwich and a side, and you can even use the bamboo lid as a cutting board.
Copyright © 1996 Farrokh Alemi, Ph.D. Created on Tuesday, September 17, 1996. Sunday, October 06, 1996 4:20:30 PM Most recent revision 09/29/2008. This page is part of a course lecture on Assessing Probability of Rare Events.