joint probability of dependent variables Consider three integer random variables $a, b$ and $c$ with common distribution function, $m(-1)=m(0)=m(1)=\displaystyle\frac{1}{3}$. 
What is the analytic way of calculating $P(ab=1, ac=-1)$ ?
Many thanks.
EDIT
$a$, $b$ and $c$ are independent variables.
EDIT2
What I mean by analytic way is a method which [possibly] avoids counting.
The reason: the non-toy version of the problem looks similar to this:
$P\left(\bigwedge \sum a_i a_j = t_k \right)$, e.g. $P(a_1a_2+a_1a_3=-1, a_1a_3+a_2a_4+a_1a_5=2, a_2a_5=0)$, 
or worse
$P\left(\bigwedge \left( \sum a_i a_j a_k = t_l, \sum a_m a_n = t_p, \sum a_q = t_r \right) \right)$.
Again all variables involved (i.e. $a_i$) are independent and identically distributed with a common distribution function, $m$.
 A: It is not clear what "analytic way" means. To follow the common convention, we use capital letters to denote random variables.
Observe that if $A$, $B$, and $C$ are possibly not independent, then we cannot calculate the required probability from the information given. For example, if $A=B=C$, then the probability is $0$. If $A=B$ and $C=-A$, then our event $AB=1$ and $AC=-1$ occurs with probability $\dfrac{2}{3}$.
Let us assume now that $A$, $B$, and $C$ are independent, as the edited version of the post specifies.  
We have $AB=1$ and $AC=-1$ if (i) $A=B=1$ and $C=-1$ or (ii) $A=B=-1$ and $C=1$. The probability of (i) is $\dfrac{1}{3^3}$, as is the probability of (ii). Add.  
A: Here's one way.  We want the probability that $a$ and $b$ have the same sign and $a$ and $c$ have opposite signs.  So it's
$$
\Pr(a\ne0\ \&\ b\text{ has the same sign as }a\ \&\ c\text{ has the opposite sign})
$$
$$
= \Pr(a\ne0)\cdot\Pr(b\text{ has that same sign}\mid a)\cdot\Pr(c\text{ has the opposite sign}\mid a,b).
$$
If they're independent and have the distributions given, the the second and third probabilities are each $1/3$.  The first is $2/3$.  If they're not independent, some knowledge of the joint distribution beyond what was given is needed.
