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

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 $2^{\omega}$.

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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.

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Mr.Stein,thank you.yes,the the unconditional probability of each is zero.but the main question is the conditional probability.I have edit the post as appending "suppose L is selected randomly under the fair-coin (Lebesgue-Cantor) measure on $2^{\omega}$. –  XL _at_China Jul 22 '11 at 3:21
Probability conditioned on measure zero events is not well-defined. I also don't understand your addition because you already mentioned the measure was the uniform measure on Cantor space. –  Noah Stein Jul 22 '11 at 11:36
I just clarify what I have said in other way Since I do not know why you said the conditional probabilities are undefined or not well-defined. I understand now why you think the conditional probabilities are undefined. –  XL _at_China Jul 22 '11 at 13:28
@Mr.Stein.then how do you think about the conditional probability of $P(F|R)$?I think $P(F|R)=1$,although $P(R)=0$? –  XL _at_China Jul 23 '11 at 5:06
Sure, it would be natural enough to define $P(F\mid R) := 1$, but that does not make it clear that the other conditional probabilities make sense. –  Noah Stein Jul 23 '11 at 9:25