### Definition(s)

#### Probability

Numerical value between zero and one assigned to a random event (which is a subset of the sample space) in such a way that the assigned number obeys three axioms:

1) the probability of the random event “A” must be equal to, or lie between, zero and one;

2) the probability that the outcome is within the sample space must equal one; and

3) the probability that the random event “A” or “B” occurs must equal the probability of the random event “A” plus the probability of the random event “B” for any two mutually exclusive events.

*Source:API STANDARD 780, Security Risk Assessment Methodology for the Petroleum and Petrochemical Industries, First Edition, May 2013. Global Standards*

#### Probability

Numerical value between zero and one assigned to a random event (which is a subset of the sample space) in such a way that the assigned number obeys three axioms: (1) the probability of the random event ―A‖ must be equal to, or lie between, zero and one; (2) the probability that the outcome is within the sample space must equal one; and (3) the probability that the random event ―A‖ or ―B‖ occurs must equal the probability of the random event ―A‖ plus the probability of the random event ―B‖ for any two mutually exclusive events.

**Sample Usage:** The probability of a coin landing on “heads” is 1/2.

**Annotation:**

- 1.Probability can be roughly interpreted as the percent chance that something will occur. For example, a weather forecaster’s estimate of a 30 percent chance of rain in the Washington, DC area is equivalent to a probability of 0.3 that rain will occur somewhere in Washington, DC.
- 2.A probability of 0 indicates the occurrence is impossible; 1 indicates that the occurrence will definitely happen.
- 3.Probability is used colloquially as a synonym for likelihood, but in statistical usage there is a clear distinction.
- 4.The probability that event A occurs is written as P(A).
- 5.Event A and event B are mutually exclusive if they cannot occur at the same time. For example, a coin toss can result in either heads or tails, but both outcomes cannot happen simultaneously.
- 6.Event A and event B are statistically independent if the occurrence of one event has no impact on the probability of the other. Examples of two events that are independent are the systems designed to prevent an attack as described the Fault Tree example and Event Tree example. The probability that the Personnel Action to Stop Attack is successful is not affected by whether the Security Equipment to Stop Attack is successful and vice versa. Two events that may not be independent are the collapse of a bridge and the occurrence of a major earthquake in the area. Clearly the probability of a bridge collapse can be affected by the occurrence of a major earthquake. However, the two events may also be independent: a bridge can survive an earthquake and a bridge can collapse in the absence of any earthquake.
- 7.Conditional probability is the probability of some event A, given the occurrence of some other event B, written as P(A|B). An example is the conditional probability of a person dying (event A), given that they contract the pandemic flu (event B).
- 8.Joint probability is the probability of two events occurring in conjunction -that is, the probability that event A and event B both occur, written as ) or P(AB) and pronounced A intersect B. The probability of someone dying from the pandemic flu is equal to the joint probability of someone contracting the flu (event A) and the flu killing them (event B). Joint probabilities are regularly used in Probabilistic Risk Assessments and Event Trees.
- 9.Conditional and joint probabilities are related by the following formula:

P(A|B) = P(AB)/P(B) (1)

If events A and B are statistically independent then

P(A|B) = P(A)

and the relationship (1) above becomes

P(A) × P(B) = P(AB)

Consequently, for statistically independent events, the joint probability of event A and event B is equal to the product of their individual probabilities. An example of the joint probability of two independent events is given in the Event Tree example. If the probability that Personnel Action to Stop Attack fails equals P(A) and the probability that Security Equipment to Stop Attack fails equals P(B) then

Probability of Successful Attack = P(AB)

= P(A) × P(B)

= 0.1 × 0.3

= 0.03

as calculated in the Event Tree example (see Figure A on page 14).

10.Marginal probability is the unconditional probability of event A, P(A). It is the probability of A regardless of whether event B did or did not occur. If B can be thought of as the event of a random variable X having a given outcome, then the marginal probability of A can be obtained by summing (or integrating, more generally) the joint probabilities over all outcomes for X.

Suppose, for example, that event A is the occurrence of an illegal person entering the country and X is the random variable of where he entered the country. Then there are two possible outcomes of X: either he entered through an official point of entry (event B), or he did not (event B’ pronounced B-not). Then the probability of the person entering the country, P(A), is equal to the sum of the joint probabilities of him entering by traveling through a point of entry plus the probability of him entering by not traveling through a point of entry. P(A) = P(AB) + P(AB’). This is called marginalization.

*Source: DHS Risk Lexicon, U.S. Department of Homeland Security, 2010 Edition. September 2010 Regulatory Guidance*

#### Probability

Measure of the chance of occurrence expressed as a number between 0 and 1, where 0 is impossibility and 1 is absolute certainty. NOTE See definition 3.6.1.1, Note 2.

*Source: ISO Guide 73:2009(E/F), Risk Management – Vocabulary, First Edition, 2009. Global Standards*