approach is to assume that all events are statistically independent. While often
true, it is sometimes patently false (and overlooked - or worse, ignored). The usual
computation of the probability of two elements NOT failing is
P(A and B) = P(A)*P(B),
where A and B represent the survival of elements A and B, both elements having to survive for the structure to survive. By extension then, for n Elements,
P(not failing) = P(A)*P(B)* ... P(En),
since all must survive to preclude structural failure.
The problem, and it is a serious one, is that this simple equation so often used by us engineers, isn't quite right. The correct equation is this:
P(A and B) = P(A)*P(B|A),
where P(B|A) is the *conditional* probability of B given that A has already occurred. This distinction is not trivial. Of course when A and B are independent, P(B|A) = P(B), by the definition of statistical independence. As an aside, if the events are perfectly correlated,
P(B|A)=1, and P(A and B) = P(A).
like to say "F=ma" as a declaration of the obvious. But we don't really
mean it - unconditionally, anyway. We know, but leave unsaid, that F=ma if (and only
if) the mass is constant, which it usually is. But what Newton really said was
F=d(mv)/dt, which is F=ma, on the implicit condition that d(m)/dt=0.
That is ...
In many situations such an
oversight would be without consequence. It would be disastrous, however, in
computations regarding rocket propulsion where the mass being accelerated diminishes with
time as it is being expended in combustion.
Nor are our statistician friends completely without sin ...