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

(1.6)

If is known, and .

(1.7)

Events and are said to be dependent if the knowledge that has occurred affects the probability that will occur, and vice versa . For example, let 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 be the draw of a card from this reduced pack of cards. Let be the event ‘ an Ace is drawn ’. Thus = 4/52 = 1/13. (Note here the conditioning information 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 will be omitted in many cases but its existence should never be forgotten.) Let be the event ‘ an Ace is drawn ’ also. Then 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 .) Thus . It is now possible to formulate the third law of probability for dependent events.

(1.8)

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

Example 1.3A 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 1.3Presence 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.

Denote the presence of DVI by and the presence of DAI by . 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 is estimated by 14/120, the total number of DVI divided by the total number of fatalities. Similarly 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, and , the third law of probability, ( 1.8) may be written as