Statistics and the Evaluation of Evidence for Forensic Scientists. Franco Taroni

      This law may be generalised to more than two events. Consider events . If they are mutually independent then

      

      1.7.10 Law of Total Probability

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.

       Law of Total Probability

      If are mutually exclusive and exhaustive events,

      (1.12)

      This is sometimes known as the extension of the conversation (Lindley 1991)

      An example for blood types and paternity cases is given by Lindley (1991). Consider two possible groups, (Rh) and (Rh+) for the father, so here . Assume the relative frequencies of the two groups are and , respectively. The child is Rh (event ) and the mother is also Rh (event ). The probability of interest is the probability a Rh mother will have a Rh child, in symbols . This probability is not easily derived directly but the derivation is fairly straightforward if the law of total probability is invoked to include the father.

(1.13)

This is a generalisation of the law to include information . If both parents are Rh, event ( and ), then the child is Rh