What to read first: Likelihood of a future event size total within a range 

Risk specialists  Version 1.0 Beta 
Previous pages described how to assess the likelihood of a future event size total within a range, using Monte Carlo trials. This page discusses some results.
As at 11 December 2018, there are some questions about the methods leading to these results, arising from community consultation on LinkedIn. This page will be updated as questions are resolved. 
There are six different cases for comparison, set out in a table below.
This table is hard work, so leave it until you have time and patience.
The case study behind the table builds on a previous post in which the ‘total’ of interest was the total amount of work time lost due to staircase injuries. An ‘event’ is a staircase injury with nonzero lost work time. The unit of time and the future prediction period are each one year. Lost work time (event size) is assumed to be measured in hours, so an event size of ‘4’ means 4 lost hours (or about half a working day).
Unlike in the previous example for simple incident counts, there is no point in separating ‘minor’ injuries from more severe injuries. The severity of each injury incident is measured by its lost working hours.
The six cases in the table represent different assumptions about the history of staircase injury incidents and the distribution of injury severity per incident, as measured in lost hours.
If you’re over staircase injuries, you can imagine the lost hours are periods of data centre downtime arising from power supply failures. Like staircase injuries, power supply failures might reasonably be assumed to occur unpredictably and with varying amounts of lost time per instance. In either application, the probability distributions are assumed to be Poisson (for event counts) and LogNormal (for event sizes).
Cases  
A  B  C  D  E  F  
Case inputs  
Median (50th percentile) event size  4  3  3  
95th percentile event size  40  50  100  
Events in history  150  5  150  5  150  5 
Duration of history  10  1/3  10  1/3  10  1/3 
Duration of future interval  1  1  1  
Case outputs  
Most likely annual total of lost hours  ~110  ~150  ~100 to ~150*  ~200 to ~300*  ~500 to ~1000*  ~500* 
Mean annual total of lost hours  ~160  ~190  ~195  ~230  ~430  ~530 
5th percentile of total lost hours (lower end of 90% prediction interval)  ~49  ~37  ~41  ~32  ~52  ~40 
95th percentile of total lost hours (upper end of 90% prediction interval)  ~350  ~450  ~450  ~610  ~1300  ~1650 
Likelihood of 0 incidents and 0 lost hours  0%  <1%  0%  ~0.05%  0%  ~0.05% 
Likelihood of 200 or more lost hours  ~25%  ~38%  ~32%  ~42%  ~59%  ~65% 
Likelihood of 1000 or more lost hours  ~0.1%  ~0.2%  ~1%  ~1.5%  ~8%  ~11% 
*Imprecise due to the constraints of the Clear Lines Excel workbook, which is limited to 4000 trials. A much larger number of trials could pin all the numbers with any desired precision.
Comments on the table of results
In cases A and B, incidents are assumed to result in a median of 4 lost work hours, and 95% of incidents are assumed to lead to less than 40 lost hours. In case A, there is long history, ten years with 150 incidents, suggesting a longterm average of 15 per year. In case B, the frequency of incidents is inferred from just four months of experience, which included five incidents, projecting to the same 15 per year. But in case B there is much more uncertainty around that average frequency.
In cases C and D, there is a small change to the assumptions about the hours lost per incident. The median lost time per incident is assumed to be 3 hours, less than for A and B. But the 95^{th} percentile for lost time is increased slightly, from 40 to 50 hours. The frequency assumptions for case C match those of A, a long history averaging 15 incidents per year. Expectations for the total lost hours in the future year are worse, despite the reduced median hours per incident, due to the seemingly small increase in the 95^{th} percentile for lost hours from an incident. In case D, the incident frequency is as for case B, inferred from five incidents in four months. The changes between B and D result in a marked increase in the most likely annual total for lost hours, and a blowout in the 95^{th} percentile for the annual total of lost hours.
Cases E and F push the 95^{th} percentile for lost hours from an incident from 50 hours to 100 hours. The effects are dramatic. Yet in the absence of data, the 95^{th} percentile of 100 hours for a single incident might seem just as reasonable as assuming that 95^{th} percentile is 40 or 50 hours. Case E assumes a good knowledge of event frequency from a long history (150 incidents in ten years). Case F infers the event frequency from a short history (five instances in four months), so introduces further uncertainty.
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