What is the conditional probability or probability of classes of languages?


Let $E,C,S,F,R $ be the class of computably enumerable languages,computable languagesl,context-sensitive anguages,context-free languages and regular languages respectively. $E$ is class of all computably enumerable languages and it's subset of $E$ is the Cantor space,take the uniform probability measure on the Cantor space,then what is the probability or conditional probability $P(C),P(S),P(F),P(R),P(S|C),P(F|C),P(R|C),\cdots,P(R|F)$ ?suppose L is selected randomly under the fair-coin (Lebesgue-Cantor) measure on $2^{\omega}$.



What is the conditional probability or probablity of classes of  languages?
The probability measure you're asking about is nonatomic, but your events of interest are countable. Therefore the unconditional probability of each is zero, and the conditional probabilities are undefined.


