STATISTICAL TOPICS AND ANALYSIS IN RISK ASSESSMENT - Risk Management Applications in Pharmaceutical and Biopharmaceutical Manufacturing - page 80

A rare event is one that is uncommon or infrequent and unusual or distinctive
in some way. Such an event occurs because the scope of possibilities is so much
greater than we can imagine. For example, there are thousands of pathways in a
computer that can lead to truly weird results.
Rare is also relative. A once in a million per day event for a small village is
happening about 300 times a day in the whole United States. Six Sigma programs
aim for defects of around four defects per million. In a tablet batch of 1.2 million
running at Six Sigma, can we really expect to find those four defective tablets?
It is also helpful to differentiate between extreme values and outliers. An
extreme value exceeds the ordinary or the usual. They are situated at the farthest
possible point from the center of the data and are still considered to be part of
the population. They have a very low probability of occurrence. For example,
heavy tablets or capsules or under-filled vials are extreme values. Floods and
natural disasters are another example. They are the result of an accumulation of
common causes.
On the other hand, outliers are so far from the rest of the population that they
are not considered to be part of the expected distribution. They are often the
result of nonrandom special causes. Examples include an empty sugar packet, an
empty vial, or a double-struck tablet. Manmade disasters are outliers. Outliers
are generally rare events.

4.9.1 Calculating POISSON

Poisson is useful in determining the likelihood of small occurrences where there
are many chances for the event to occur.
When calculating probabilities of occurrence of hazards such as lack of sterility
or the presence of endotoxin, the law of small numbers ( or the law of rare events )
is applicable. Poisson distribution describes the law of rare events and is shown
mathematically as

P (X) = λ X e −λ
X !

where:

X = the number of expected occurrences

e = base of the natural log

average number of occurrences of the event
For example, if we wanted to determine the probability of having 0 steril-
ity failures in the upcoming year for a manufacturing site that has a historical
average of 2, we could use Poisson. In this example, X , the number of expected
occurrences, is 0, and

λ =

λ , average number of occurrences of the event, is 2.

So:

X = 0

λ = 2