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Conditional probability
Although everybody knows the formula of conditional probability,
it is useful to derive it here^{3.5}.
The notation is
,
to be read ``probability of given '', where stands for
hypothesis.
This means: the probability that will occur under the
hypothesis that has occurred^{3.6}.
The event can have three values:
 TRUE:
 if is TRUE and is TRUE;
 FALSE:
 if is FALSE and is TRUE;
 UNDETERMINED:
 if is FALSE; in this case we are merely
uninterested in what happens to . In terms
of betting, the bet is
invalidated and none loses or gains.
Then can be written
,
to state explicitly that it is the probability of
whatever happens to the rest of the world
( means all possible events). We realize immediately
that this condition is really too vague and nobody would
bet a penny on a such a statement. The reason for usually
writing
is that many conditions
are implicitly, and reasonably, assumed in most
circumstances. In the classical problems of coins and dice, for example,
one assumes that they are regular. In the example
of the energy loss,
it was implicit (``obvious'') that the
high voltage was on (at which voltage?)
and that HERA was running (under which condition?).
But one has to take care: many riddles are
based on the fact that one tries to find a solution which is
valid under more strict conditions than those explicitly stated
in the question, and many people make bad business deals by signing
contracts in which what ``was obvious''
was not explicitly stated.
In order to derive the formula of conditional probability
let us assume for a moment that it is reasonable to
talk about
``absolute probability''
,
and let us rewrite
where the result has been achieved through the following steps:
 (a)
 implies (i.e.
)
and hence
;
 (b)
 the complementary events and
make a finite partition of ,
i.e.
;
 (c)
 distributive property;
 (d)
 axiom 3.
The final result of () is very simple:
is equal to the probability that occurs and also
occurs, plus the probability that occurs but does not
occur. To obtain
we just get rid of the subset of
which does not contain (i.e.
)
and renormalize the probability
dividing by , assumed to be different from zero. This guarantees
that if then
.
We get, finally, the well known formula

(3.2) 
In the most general (and realistic)
case, where both and are conditioned by the occurrence of
a third event , the formula becomes

(3.3) 
Usually we shall make use of ()
(which means
) assuming that has been
properly chosen.
We should also remember that () can be resolved
with respect to
, obtaining the well known

(3.4) 
and by symmetry

(3.5) 
We remind that two events are called independent if

(3.6) 
This is equivalent to saying that
and
,
i.e. the knowledge that one event has occurred does not change the
probability of the other. If
then the events
and are correlated. In particular:
 if
then and are positively correlated;
 if
then and are negatively correlated.
Next: Bayes' theorem
Up: Conditional probability and Bayes'
Previous: Dependence of the probability
Contents
Giulio D'Agostini
20030515