Probability of expansions

Leo Marcus leomarcus2 at
Thu Jul 23 20:24:15 EDT 2020

Is this known?

Given a first-order theory T in language L and a first-order theory T'
containing T in language L' containing L, what is the probability a finite
model of T has an expansion to a model of T'?  I.e., what is the ratio
between those finite models of T that do and those that don't? How about
for those models of a fixed cardinality? When does  the limit as the
cardinality tends to infinity exist?
Is there any restriction on what Prob (Exp(T, T')) can be as T and T' vary?

Any references would be appreciated.
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