The conditional
probability
of A given B is represented by P(A|B). The variables A and B are said
to be *independent*
if P(A)= P(A|B) (or alternatively if P(A,B)=P(A) P(B) because of the formula
for conditional probability ).

**Example****1
**Suppose
Norman and Martin each toss separate coins. Let A represent the
variable "Norman's toss outcome", and B represent the
variable "Martin's toss outcome". Both A and B have two
possible values (Heads and Tails). It would be uncontroversial to
assume that A and B are independent. Evidence about B will not change
our belief in A.

**Example****2
**Now
suppose both Martin and Norman toss the same coin. Again let A
represent the variable "Norman's toss outcome", and B
represent the variable "Martin's toss outcome". Assume also
that there is a possibility that the coin in biased towards heads but
we do not know this for certain. In this case A and B are not
independent. For example, observing that B is Heads causes us to
increase our belief in A being Heads (in other words P(a|b)>P(b)
in the case when a=Heads and b=Heads).

In Example 2
the variables A and B are both dependent on a separate variable C,
"the coin is biased towards Heads" (which has the values
True or False). Although A and B are not independent, it turns out
that once we *know*
for certain the value of C then any evidence about B cannot change
our belief about A. Specifically:

P(A|C) = P(A|B,C)

In such case we
say that *A
and B are conditionally independent given C*.

In many real life situations variables which are believed to be independent are actually only independent conditional on some other variable.

**Example
3**
Suppose that Norman and Martin live on opposite sides of the City and
come to work by completely different means, say Norman comes by train
while Martin drives. Let A represent the variable "Norman
late" (which has values true or false) and similarly let B
represent the variable "Martin late". It would be tempting
in these circumstances to assume that A and B must be independent.
However, even if Norman and Martin lived and worked in different
countries there may be factors (such as an international fuel
shortage) which could mean that A and B are not independent. In
practice any model of uncertainty should take account of all *reasonable*
factors. Thus while, say, a meteorite hitting the Earth might be
reasonably excluded it does not seem reasonable to exclude the fact
that both A and B may be affected by a Train strike (C). Clearly P(A)
will increase if C is true; but P(B) will also increase because of
extra traffic on the roads. Thus the situation is represented in the
following animation (which is actually a BBN)

Now, "Martin late" and "Norman late" are conditionally independent given "Train strike":

See also the section on transmitting evidence in BBNs.