The probability P(F) represents the searched ecodesign function; a function that has multiple variables and lies between the values of 0 and 1, and is defined by f(Q=xi, H=yi, E=Zi). However, according to the nature of sets Q, H and E (independent or dependent) and the rules of multiplication in the probability theory, two possible cases (Figures 2a and 2b) may be presented; these are:
a. The sets Q, H, E are independent (Figure 2a): a condition which does not satisfy the searched objective because the probability of the intersection is an empty set. Thus, it may be expressed by the following equation:
f(Q=Xi, H=y, E=z) = P(F ) = P(0) = 0 (5)
b. The sets Q, H, E are dependent (Figure 2b and/or 2c): a condition that does satisfy the searched objective. Thus, the probability of the occurred intersection can be written as:
f(Q=Xi, H=y, E=Zi) = P(F ) = P(Q) x Pq (H) x Pq^h (E) (6)
where, P(Q) represents the probability of an achievable quality;
PQ (H) represents the probability of an achievable health, knowing that quality has been achieved; and
PQnH (E) represents the probability of an achievable environment, knowing that quality and health have been achieved.
The following probability notations may be of some use in a certain literature reviews:
Pq (H) = P(H |Q) = P(QrH)/P(Q) (7a)
PQrH (E) = P(E IQrH)=P(QrHrE)/P(QrH) (7b)
Equations (7a) and (7b) are only valid when P(Q) and P(QrH) are strictly greater than zero. On the other hand and according to the probability analysis, the realization of the event F can take several values as a final result. This latter can be recapitulated by the following possible events (Attaf, 2007):
• If probability value is null (P(F ) = 0), then the event is impossible.
• If probability value is equal to 1 (P(F ) = 1), then the event is certain.
• If probability value is located between the two extreme values (0<P(F )<1), then it does exist a series of probable events.
