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

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one of those values. Let images be any event and let images denote the probability that images occurs. Then images. For an event that is known to be impossible, the probability is zero. Thus if images is impossible, images. For an event that is known to be certain, the probability is one. Thus, if images is certain, images. This law is sometimes known as the convexity rule (Lindley 1991).

      Consider the hypothetical example of the balls in the urn of which a proportion images are black and a proportion images white, with no other colours present, such that images. Proportions lie between 0 and 1; hence images. For any event images. Consider images, the drawing of a black ball. If there are no black balls in the urn, this event is impossible then images. This law is sometimes strengthened to say that a probability can only be 0 when the associated event is known to be impossible.

      The first law concerns only one event. The next two laws, sometimes known as the second and third laws of probability, are concerned with combinations of events. Events combine in two ways. Let images and images be two events. One form of combination is to consider the event ‘images and images’, the event that occurs if and only if images and images both occur, sometimes denoted images. This is known as the conjunction of images and images.

      Consider the roll of a six‐sided fair die. Let images denote the throwing of an odd number. Let images denote the throwing of a number greater than 3 (i.e. a 4, 5, or 6). Then the event ‘images and images’ denotes the throwing of a 5.

      The second form of combination is to consider the event ‘images or images’, the event that occurs if images or images (or both) occurs. This is known as the disjunction of images and images.

      Consider again the roll of a single six‐sided fair die. Let images, the throwing of an odd number (1, 3, or 5), and images, the throwing of a number greater than 3 (4, 5, or 6), be as before. Then ‘images or images’ denotes the throwing of any number other than a 2 (which is both even and less than 3).

      Secondly, consider drawing a card from a well‐shuffled pack of 52 playing cards, such that each card is equally likely to be drawn. Let images denote the event that the card drawn is a spade. Let images denote the event that the card drawn is a club. Then the event ‘images or images’ is the event that the card drawn is from a black suit.

       Second Law of Probability

      The second law of probability concerns the disjunction ‘images or images’ of two events. Events are called mutually exclusive when the occurrence of one excludes the occurrence of the other. For such events, the conjunction ‘images and images’ is impossible. Thus images and images.