The main diagonal elements commonprob[i,i] are interpreted as
probabilities \(p_{A_i}\) that a binary variable \(A_i\)
equals 1. The
off-diagonal elements commonprob[i,j] are the probabilities
\(p_{A_iA_j}\) that both \(A_i\) and \(A_j\) are 1.
This programs checks some necessary conditions on these probabilities
which must be fulfilled in order that a joint distribution of the
\(A_i\) with the given probabilities can exist.
The conditions checked are
$$0 \leq p_{A_i} \leq 1$$
$$\max(0, p_{A_i} + p_{A_j} - 1) \leq p_{A_iA_j} \leq
\min(p_{A_i}, p_{A_j}), i \neq j$$
$$p_{A_i} + p_{A_j} + p_{A_k} - p_{A_iA_j} -p_{A_iA_k} - p_{A_jA_k}
\leq 1, i \neq j, i \neq k, j \neq k$$