Top Tips for Probability Analysis

Read on to learn some top tips on how to make sure you are defining and using probability values correctly within a safety or reliability assessment.

1. DO Define Probability Correctly

   In particular:
   a. Probabilities must lie in the range 0 to 1 (where 0 represents an event which can never occur and 1 represents a definite certainty)
   b. Probabilities are defined as proportions and are therefore dimensionless quantities, i.e. they cannot be expressed in terms such as “per hour”.

2. DON’T Use Probability to describe a Rate of Occurrence

   Sometimes people use the phrase “the probability is 0.27 per year” to mean “the rate of occurrence is 0.27 per year”. It is very common in our experience to find the use of a probability statement to describe a rate of occurrence. Unfortunately this can lead to errors when calculating the probability of two or more independent events occurring within a specified interval (see example below).

3. DON’T Apply the Laws of Probability to Rates of Occurrence

   Consider two independent events A and B, both of which have a rate of occurrence but are incorrectly described as having “a probability of 0.27 per year” for A and “a probability of 0.85 per year” for B. Using the laws of probability, this would imply that

   P(A OR B) = P(A) + P(B) – P(A)×P(B) = 0.27 + 0.85 – 0.27×0.85 = 0.8905

   However because 0.27 and 0.85 are rates of occurrence rather than probabilities, the rate at which either A or B occurs is simply 0.27+0.85=1.12 times per year (a “probability” greater than 1). The Laws of Probability simply do not apply to rates!

4. DO Convert Rates to Probabilities before Applying the Laws of Probability

   The probability P of an item having a constant failure rate ‘λ’ failing within a specified time interval ‘t’ can be calculated using the formula:

   P = 1-exp(-λt)

   Therefore, it is a straightforward matter to calculate the probability of failure for an item within the time interval ‘t’ of interest.

   NOTE: P ≈ λt for small values of λt but this is still a dimensionless quantity

   Incorrect application of the laws of probability is very common, and every effort should be made both by practitioners and managers to avoid this as far as possible. To help with this, we offer the following recommendations:

To Practitioners

• Define probabilities correctly, making sure that they are always dimensionless quantities.
• Avoid the temptation to take shortcuts – do not apply the laws of probability to quantities that are not strictly probabilities.
• In written reports avoid the use of expressions such as “the probability is 0.27 per year” which imply that probability is not a dimensionless quantity.

To Managers

• Insist that probabilities defined in reports produced within your organisation comply with the “dimensionless” quality of a true probability – this imposes a discipline that will help to ensure that any probability analysis is carried out correctly.
• Ensure that probabilities have been correctly defined and analysed in all reports that consider safety issues or that are important to your business.