Читать книгу Statistics and the Evaluation of Evidence for Forensic Scientists - Franco Taroni - Страница 42

Events are said to be mutually exclusive and exhaustive if one of them has to be true and only one of them can be true; they exhaust the possibilities and the occurrence of one excludes the possibility of any other. Alternatively, they are called a partition. The event formed from the conjunction of the individual events is certain to happen since the events are exhaustive and exclusive. Thus, it has probability 1 and

(1.11)

a generalisation of the second law of probability, (1.7), for exclusive events. Consider as an example allelic distributions at a locus, e.g. locus TPOX. There are five alleles, , and 12, and these are mutually exclusive and exhaustive.

Consider for events and . Let be any other event. The events ‘ and ’ and ‘ and ’ are exclusive. They cannot both occur. The event “‘ and ’ or ‘ and ’ ” is simply . For example, let be male, be female, be left‐handed. Then

 ‘’ denotes a left‐handed male,

 ‘’ denotes a left‐handed female.

The event ‘ “ and ” or “ and ” ’ is the event that a person is a left‐handed male or a left‐handed female, which implies the person is left‐handed (). Thus,

The argument extends to any number of mutually exclusive and exhaustive events to give the law of total probability.