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(1.8)

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 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

(1.9)

where the conditioning on has been omitted.