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

Statistics and the Evaluation of Evidence for Forensic Scientists

evidence that the PoI was seen at the scene of the crime but this evidence may be felt to be unreliable. Its value will then be lessened.

The value of scientific evidence will be conditioned on the background data relevant to the type of evidence being assessed. Evidence concerning frequencies of different DNA profiles will be conditioned on information regarding ethnicity of the people concerned for the values of these frequencies. Evidence concerning distributions of the refractive indices of glass fragments will be conditioned on information regarding the type of glass from which the fragments have come (e.g. building window, car headlights etc.). The existence of such conditioning events will not always be stated explicitly. However, they should not be forgotten. As stated above, all probabilities may be thought of as conditional probabilities. The first two laws of probability can be stated in the new notation, for events images and information images as:

First law of probability for dependent events

(1.6) equation

If images is known, images and images .

Second law of probability for dependent events

(1.7) equation

Events images and images are said to be dependent if the knowledge that images has occurred affects the probability that images will occur, and vice versa. For example, let images be the outcome of a draw of a card from a well‐shuffled pack of 52 playing cards. This card is not replaced in the pack so there are now only 51 cards in the pack. Let images be the draw of a card from this reduced pack of cards. Let images be the event ‘an Ace is drawn’. Thus images = 4/52 = 1/13. (Note here the conditioning information images that the pack is well‐shuffled, with its implication that each of the 52 cards is equally likely to be drawn has been omitted for simplicity of notation; explicit mention of images will be omitted in many cases but its existence should never be forgotten.) Let images be the event ‘an Ace is drawn’ also. Then images is the probability that an Ace was drawn at the second draw, given that an Ace was drawn at the first draw (and given everything else that is known, in particular that the first card was not replaced). There are 51 cards at the time of the second draw of which 3 are Aces. (Remember that an Ace was drawn the first time which is the information contained in images .) Thus images . It is now possible to formulate the third law of probability for dependent events.

Third law of probability for dependent events

(1.8) equation

Thus in the example of the drawing of the Aces from the pack, the probability of drawing two Aces is

Example 1.3 A study of the brains of 120 road accident fatalities given in Pittella and Gusmäo (2003, Table 2), reproduced in Lucy (2005) observed the numbers of diffuse vascular injuries (DVI) and diffuse axonal injuries (DAI) with the results presented in Table 1.3.

Table 1.3 Presence and absence of diffuse vascular injuries (DVI) and diffuse axonal injuries (DAI) in 120 road accident fatalities.

Source: From Pittella and Gusmäo (2003). ©ASTM International. Reprinted with permissions of ASTM International.

	DAI
DVI	Present	Absent	Total
Present	14	0	14
Absent	82	24	106
Total	96	24	120

Denote the presence of DVI by images and the presence of DAI by images . Then various probabilities for the incidences of the two types of injuries in the population of road accident fatalities can be estimated from this sample of 120 fatalities. Thus images is estimated by 14/120, the total number of DVI divided by the total number of fatalities. Similarly images is estimated by 96/120, the number of DAI divided by the total number of fatalities.

The third law of probability for dependent events (1.8) can be verified using Table 1.3. For example,

Alternatively

Thus, for dependent events, images and images , the third law of probability, (1.8)

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