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Topic 35: A different approach to the basic R = P x C formula

Foivos Theofilopoulos's picture

I was first introduced to risk assessment during my first degree, as part of a module on Human Factors. I have to make the disclaimer that we only touched semi-quantitative methods of risk assessment; we did not delve into the fully quantitative.
One thing that I find really interesting and debate-worthy is the use of one more variable in the familiar  R = P x C  formula.  Our textbooks were using a slightly different approach, adding the variable E (for exposure) in the formula, making the equation  R = P x E x C . The exposure factor comes into consideration when you have events and hazards manifesting non-continuously, but at certain times throughout the year, at specific points of day etc.

 I will once more state that I'm refering to semi-quantitative methods, so instead of distribution functions of continuous random variables we are dealing with discrete rand. var., so imagine the end result as a matrix, with exposure being assigned different values according to statistical tables. So, do you thing that this approach is more complicated than the one we are being taught in our course? Maybe it can be applied only to certain areas/specific problems?


Foivos Theofilopoulos's picture

For a very basic example, imagine a component that is susceptible to ice (for whatever reason, it does not matter in this example), so it is given a 30% chance of failing when covered in ice. According to the first formula, we just put 30% in the Probability variable and we’re done. But how does the time of year affect the equation? Do you have the same risk during summer and winter? Does someone have to calculate an entire set of conditional probabilities (Prob. of failure given snow in summer,  Prob. of failure given snow in winter etc etc)?
In this mindset, the variable E helps distinguish between the industry-tested chance of failure of a component (due to a particular cause) and the various conditions that can expose the component to a hazard.

RossWinter's picture

I like the idea of adding the extra component (E) into the equation because it helps create a more accurate risk level. The original equation (R = P x C) is good for having a rough estimation of the risk involved in a project or technique, however this extra dimension may seem a small addition but it does help focus the rough figure in, without much extra calculation. the technique is still simple so other, more accurate calculations will have to be undertaken if a more detailed prediction is required, but the R = P x C x E will do for the basics.


Ross Winter Msc Renewable Energy

Soseleye F. Ideriah's picture

Relating risk and exposure is an interesting method of risk analysis. It makes sense that a hazard poses no threat if there is no exposure to the hazard. Also, a change in the level of exposure to a hazard would change the probability of a failure event occurring. This brings me to my argument.

I am of the opinion that the probability parameter used in the risk defining equation (Risk = Probability of failure event x Undesired consequence) is a function of exposure. In the example used by Folvos, different probabilities of failure would have to be defined for winter and for summer. The risk equation holds true as long as the correct values are used for probability and consequence.

Various factors (exposure being just one of them) work together in defining the probability of a failure event. Engineers must ensure that in analysing risk, proper care is taken to correctly define each parameter used. All factors that affect each parameter must be taken into consideration.

Andrew Strachan's picture

I have no experience with the application of the equation R = PxCxE however I would have to agree with Soseleye, if there is increased likelihood due to increased exposure then this will be reflected in the value for probability which is derived from the probability density function or hazard rate based on real data.

In terms of your example, in my opinion, the length of the period of interest is fundamental in the approach taken in analysing the reliability of the system (and hence the risk). Over a period of many years it could be said that the statistical affects of higher probabilities in winter than in summer will be less important provided the data on which the probability distribution is based accounts for this.

Uko Bassey's picture

I share the same view with Soseleye in his position that the probability component in the definition of risk given by the formula R = P x C is a function of time; also Andrew has further emphasized this using probability density function hazard rate which incorporates time in the analysis.

The introduction of quantity E (exposure) in the risk equation to give, R = P X C X E as stated by Foivos looks complicated and possibly a double factor in the equation. Considering the fact that any change (increase or decrease) in the exposure of the hazard translates to a corresponding change in the probability of occurrence and hence the value of risk. Therefore, exposure time (E) is not necessary in this.

However, we might have to look critically analyze this to determine any situation that best suits the new definition while the former continues to apply as it has gained general acceptance in application not just the field of engineering.

I share the same idea with Andrew. If you read note 7 of university lectures, you can see that hazard rate is defined. hazard rate is time-dependant. It is noted in it that the failure of one component may differ from time to time. Actually, in "Risk=Probability of risk happening * Consequences of risk" formula, the probability value includes the time. I mean that the chance of failing in winter and summer are different and is not 30% for both.In other words, the value of "probability of risk happening" is a function of many factors like time.So, in summer it takes longer for the system to fail in comparison with summer. As a result, time factor will affect the P and R as well. So, the risk in winter is higher than in summer.

mohamed.elkiki's picture

Completely agree with Andrew and nina. the equation already include the exposure but in none direct way. However, the main issue is that some risks can happen and not be included due to the equation include only probability of an event failure but the risks that we face in the industry are bigger than that but we must keep in mind that all this equations and methods are ways to just decrease the risk and not eliminate it and it also mentioned in the lecture about legislation that laws will not penalize company because they caused accident but will penalize them if they didn't take precautions. Also, in Shell lecture about event tree it was very clear that equation of risk include everything about exposure.

Hanifah N. Lubega's picture

Topic looks difficult to interpret but am glad there are people who actually gave this a thought. In my opinion, probably similar in some way, the Exposure component is very important especially when analysing ergonomic risks. You might find that exposing an employee to certain emissions, radiations or heat may not have a major impact in a short time period of exposure but as they stay longer, they may be affected in one way or the other. So I think in this case we have to introduce the time of exposure aspect when carrying out risk analysis.  When related to system components it becomes very complex because it means we will have different probabilities in the different times or seasons and this implies that if we are to improve reliability we need to factor in the behavioural change or characteristics of the component. I agree with Nina that the value of "probability of risk happening" is a function of many factors but in this case the Exposure according to my understanding is the rate at which factors that would cause failure/damage like pressure, temperature etc,  are subjected  onto the component or Worker. Foivos guide me on this.

Brenda Amanda's picture

In my opinion, it does make sense
to factor in exposure in the risk assessment calculation. If you think about it, analysing risks and
trying to mitigate them will depend on the degree of exposure to that
particular risk.

The students doing the project
management course will notice that we are in fact incorporating an exposure
factor while doing risk management of our different projects. In preparation of
a risk matrix, the relative level of exposure to different risks is integrated to
give weights and therefore help whoever is assessing the weights come up with
mitigation measures. This is especially useful when probabilities of occurrence
are not very accurate or have been presented in non-numeric values. The risk
matrix then gives an indication of how best the different risks can be managed.

Kyle McFarlane's picture

I agree with Uko the formula already factors in enough relevant information and I dont feel familair enough with the material to try and change an equation especially not one used so successfully. It seems that hazard rate takes into account what you are looking for.


However exposure is an interesting thing to debate. If no one is exposed to a proposed risk or hazard is it acceptable?

I would say no,  think of the lecture we had involving the hammer on the ledge, the fact that the hammer didnt fall and harm anyone isnt important it is the actual risk itself that is the issue regardless of exposure. 




I may be missing something here and please correct me if i
am but I feel Risk should always be measured at the highest possible value.

If a piece of equipment has a 0.5% chance of failing in
normal circumstance, but when exposed to temperatures of 57ºC it then raises
the chance of failing to 1.5%. Surely the risk is measure by the worst possible
probability.  Even if this piece of
equipment was only exposed to these temperature for 2 weeks out of 52 that
would still be the worst probability over the whole year and should be the
chance of failure used.

I am sure it the risk was to be assessed over a certain period
which it was not exposed then the lower value could be used but otherwise I
fail to see good reasoning behind using exposure, besides trying to mathematically
trying to reduce risk rather than physically?

Do you agree with me? Can you convince me otherwise?


Liam Slaven

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Samuel Bamkefa's picture

I am drawn to this topic. I have done some research on this and I have come up with the following
I think there is some sense in the R=PxExC formula (called the Kinnery method). The conventional R=PxC formula is called the RAG fomulation.

Using a simple example, I will refer to a case of the risk associated with a pump failing when it is operated. The risk associated when the pump is operated one time will be different from when it is operated 100 times. This will be of more importance is the probability of failure is time or usage varying.

Can the exposure be incorporated in the probability? Yes, it is possible. That is when the probability in the RAG formulation would have incorporated the risk. It should be made clear then that the probability we are considering in our normal formulation is not just the conventional probability as we know it.

The Kinney formulation is not meant for the trash. I think it only puts more detail and understanding into the RAG formulation


Borut Matkovic, 2012, "Market Surveillance, Training of Risk Assessment", Development and Implementation of Trade Policies and Regulations. Available on Accessed on 11/12/2012
Samuel Bamkefa

Deinyefa S. Ebikeme's picture

I will like to add to Soseleye
Ideriah and Nina Yari’s comments.

The probability of failure event
in the risk formula, R = P × C addresses two scenario. The probability of failure
could be independent of time and also time dependent (Exposure time). Time
dependent probability most times results in mechanical deterioration which leads
to the need for the reliability assessment of the components of such system.

Classical Reliability theory
helps in categorizing these failures and gives us a view of how safety, risk
and reliability interrelates in term of Hazard rate, Concept of Expected Life or Mean Time To Failure (MTTF) etc.

Deinyefa Stephen Ebikeme

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