Re: Why x+y-xy is called probabilistic OR?

Scott Ferson (scott@ramas.com)
Tue, 24 Mar 1998 00:25:23 +0100 (MET)

Several people already addressed this question, but I'll take
a stab at it. It's called the "probabilitistic OR" because if x is
the probability that event X occurs, and y is the probability
that event Y occurs, and the two events X and Y are
known to be independent, then
x OR y = x + y - (x AND y) = x + y - xy
gives the probability that at least one of the events X or Y
will occur. It's only because independence is so widely
assumed in probability theory that this function took on the
moniker of probabilistic sum. In fact if the events aren't
independent, then this function does not give the correct
probability of the disjunction (X or Y). In particular, it is
well known that the probability of at least one of the two
events occurring must lie somewhere in the interval
[ max(x, y), min(1, x+y) ]
no matter what dependence exists between the two events.
These are the classical Frechet bounds which are best
possible bounds given no information about dependence,
and which include the probabilistic sum as a special case.
Most (although not all) of the fuzzy OR operators that have
been proposed also lie within this interval.

I should mention that both the probabilistic OR and
the generalized (Frechet) OR are both correctly dual to
their respective AND operators for probabilities, contra
the suggestion in the original posting below.

Scott Ferson <scott@ramas.com>
Applied Biomathematics, 516-751-4350, fax -3435

> > Could anyone explain a connection between the operation
> > x+y-xy and probabilities. That is, the question is
> > - Why this operation is referred to as probabilistic OR, what is the
>
> > justification of this name, and what is the connection with
> probabilistic
> > theories? I am interested why it is just x+y-xy that is used for
> > probabilistic OR, and not x+y, max(x,y) or something else.
> > I need this since I have formal difficulties in applying it as
> > probabilstic OR in my reasoning, it does not work as it should,
> > there are some inconsistencies. In particular, it has to be formally
>
> > dual to the probabilistic AND operation.
>