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##

Probability of simple propositions

Let us start by recalling the basic rules of probability for propositions or hypotheses.
Let and be propositions,
which can take on only two values, for example, true or false.
The notation stands for the probability that is true.
The elementary rules of probability for simple propositions are

where means *tautology* (a proposition that is
certainly true).
The construct is true only when both and are true (logical AND),
while is true when at least one of the
two propositions is true (logical OR). is also
written simply as `' or , and is also called a *logical product*,
while is also called a *logical sum*.
is called the joint probability of and .
is the probability of
under that condition that is true. We often read it
simply as ``the probability of A, given .'' .
Equation (4) shows that the joint probability of
two events can be decomposed into conditional probabilities
in different two ways.
Either of these ways is called the *product rule*.
If the status of does not change the probability
of , and the other way around, then
and are said to be *independent*, *probabilistically*
independent to be precise. In that case,
, and
, which, when inserted in Eq. (4), yields

Equations (1)-(4) logically lead to
other rules which form the body of probability theory.
For example,
indicating the *negation* (or *opposite*)
of with , clearly
is a tautology (
),
and
is a contradiction
(
).
The symbol stands for contradiction
(a proposition that is certainly false).
Hence, we obtain from Eqs. (2) and (3)

which says that proposition is either true or not true.

** Next:** Probability of complete classes
** Up:** Rules of probability
** Previous:** Rules of probability
Giulio D'Agostini
2003-05-13