STATISTICAL TOPICS AND ANALYSIS IN RISK ASSESSMENT - Risk Management Applications in Pharmaceutical and Biopharmaceutical Manufacturing - page 78

4.7.2 Conditional Probability

A good way to define conditional probability is to think about it pictorially as
in Figure 4.1.Conditional Probability is defined as

The probability of event Failure occurring given that event Hazard has already
occurred. or “F given H”.

It is written as:

P ( Failure | Hazard ) = P (

Failure | Hazard )

P ( Hazard )

where
The vertical bar is the symbol for “given.”
This traditional definition does not really highlight the fact that we are reducing
or sectioning the sample space we wish to assess. Figure 4.2 shows two different
sets within the process and clearly shows how the reduction of the sample space
looks when compared to Figure 4.1. Set 1 is the process or product population
we are trying to describe in the risk assessment. Within this set, two subsets are
used to determine the conditional probability. The first subset (i) is simply the
magnitude of a hazard that has occurred within the population. The second subset
(ii) shows the magnitude of a specific harm that has occurred within subset (i)
Visually, Figure 4.2 shows the magnitude of a failure or harm that occurs when
the hazard is present.
Another way to say conditional probability could be “Failure within Hazard”
rather than “Failure given Hazard.”

Thus, the probability is P ( Failure | Hazard ) = P (

| )

Failure Hazard

P ( Hazard )

Multiplication Rule or Joint Probability

Set 1 - Population

Subset i - This is the hazard,
P(H).

Subset ii - This is the failure,
P(F&H)

Figure 4.2 Conditional probability described as a subset. ( See insert for color represen-
tation of the figure .)