A stochastic context-free grammar (SCFG) extends the standard context-free formalism by adding probabilities to each production:
where the rule probability p is usually written as 
.
This notation to some extent hides the fact that p
is a conditional probability, of production 
being chosen,
given that X is up for expansion.
The probabilities of all rules with the same nonterminal X on the LHS must
therefore sum to unity.
Context-freeness in a probabilistic setting translates into conditional
independence of rule choices.
As a result, complete derivations have joint probabilities that are simply
the products of the rule probabilities involved.
The probabilities of interest mentioned in Section 1 can now be defined formally.
In the following, we assume that the probabilities in a SCFG are
proper and consistent as defined in Booth:73,
and that the grammar contains no useless nonterminals (ones that
can never appear in a derivation).
These restrictions ensure that all nonterminals define
probability measures over strings, i.e., 
is a proper
distribution over x for all X.
Formal definitions of these conditions are given in
Appendix A.
