If $G$ is a finite group can and $H$ is subgroup of $G$ then is it possible to take $G$ as sample space and $H$ as an event of $G$? My work as $G$ is a group that means it is an algebraic structure $(G,*)$ where $G$ is a finite set and $*$ is an operation. And we know in probability sample space $S$ is a set of all outcome of a random experiment. So both $G$ and $S$ are finite set. So is it possible to say $G$ is a sample space. And $H$ is subgroup of $G$ so $H$ must be subset of $G$. We know event is subset of sample space. So is it possible to say subgroup $H$ as an event?
 A: It is perfectly reasonable to consider $G$ as a sample space, and ask for the probability that a randomly chosen element lies in $H \leq G$.
To respond to the implicit question in the comments, there are indeed cases where this line of reasoning is interesting. As an example, let $H = Z$ be the center of $G$. That is, $Z = \{ x \in G ~|~ \forall y \in G . xy = yx \}$. It is a theorem (due to Erdos and Turan, iirc) that if $\Pr[xy = yx] > 5/8$ then $G$ is already abelian. 
In light of this theorem, it seems natural to ask what we can say if we know, for example, $\Pr[x \in Z] > 5/8$ instead.

As for the followup question in the comments, it is also possible to use $H$ for conditional probabilities. As an example, if $K$ and $H$ are subgroups of $G$, then it is reasonable to ask $\Pr[x \in K ~|~ x \in H]$. I don't have any snappy justification for why this might be an interesting question, but I'm sure it's something somebody has considered.
In general, a group is a set with bonus properties. Any probabilistic question you might ask of a finite set is also a question you can ask of a finite group.

I hope this helps ^_^
