Suppose
that our control system X is made up of two subsystems. Let *A*
be the number of critical faults in the first subsystem and let *B*
be the number of critical faults in the second subsystem.

Suppose that

*A*
= {*a*1,*a*2,*a*3}
where *a*1=0,
*a*2=1,
*a*3=">1"

*B*
= {*b*1,*b*2,*b*3}
where *b*1=0,
*b*2=1,
*b*3=">1"

If we are
interested in the overall number of critical faults in the system,
then we speak about the joint event *A
and B*.
We write the probability of this event as

P(*A*,*B*)

P(*A*,*B*)
is called the *joint
probability distribution*
of *A*
and *B*.
Specifically, P(*A*,*B*)
is the set of probabilities:

{P(*a*1,* b*1),
P(*a*1,* b*2),
P(*a*1,* b*3),
P(*a*2,* b*1),
P(*a*2,* b*2),
P(*a*2,* b*3),
P(*a*3,* b*1),
P(*a*3,* b*2),
P(*a*3,* b*3)}

where for any *i*
and *j*,
P(*a*i,*b*j)
is the probability of the event *a*i
and *b*j.

In general, if *A*
and *B*
are variables with possible states {*a*1,
*a*2,
..., *a*n}
and {*b*1,
&ldots;, *b*m}
respectively then the joint probability distribution P(*A*,*B*)
is the set of probabilities

{ P(*a*i,*b*j)
| *i*=1,&ldots;,*n**j*=1,&ldots;,*m*}

If we
know the joint probability distribution P(*A*,*B*)
then we can calculate P(*A*)
by a formula (called *marginalisation*)
which comes straight from the third axiom, namely:

This is because
the events (*a*,*b*1),
(*a*,*b*2),
..., (*a*,*b*m)
are mutually exclusive. When we calculate P(*A*)
in this way from the joint probability distribution we say that the
variable *B*
is marginalised out of P(*A*,*B*).
It is a very useful technique because in many situations it may be
easier to calculate P(*A*)
from P(*A*,*B*).