What space to use? My apology if this question is not mathematical. I have heard of many spaces, Hilbert space, Banach space etc. But could not connect a specific problem to a space. For example if I ask a mathematical question is it automatically determines which space we are at? For example if I ask what is the probability of getting a 3 when I throw a dice of 6 faces? We know the answer to this question but while solving in which space we worked on? Is this a valid question? Another related question: is there a question which cannot be solved in Hilbert space but can be solved in Banach space or vice versa? I feel the concept of space is essential to solve any problem but cannot associate it to a specific problem.
 A: Something to keep in mind is that "Hilbert space" and "Banach space" are types of space, not spaces in themselves. Think of them as analogous to types of topological space, like Hausdorff, T2, discrete, that sort of thing.
The type of space that you use emerges naturally from the problem you're considering and the sorts of tools you have available to apply to it. When you're studying a differential equation like $u_{xx} + u_{yy} = f$ (with certain boundary conditions), for instance, then it's useful to recast the equation as a linear algebra problem, $\Delta:??\to??$, and look for a suitable "inverse." 
So you ask, what is a reasonable norm for functions I'm considering here? You want them to be twice-differentiable, so you ask for the $L^2$ norms of their first two derivatives to be finite. The completion of compactly supported functions with respect to this norm is the second $L^2$ Sobolev space. Proceeding in this way, once you have a domain and a range, you can use the tools of linear algebra to look for solutions to your problem.
This is how all the different function spaces occur "in nature." When you study the basic theory of Hilbert and Banach spaces, you're gathering experience with a class of tools that will help you deal with the spaces that arise naturally in other areas of mathematics.
A: For your first question, we're working in a discrete probability space, if you've studied measure-theoretic probability theory, you've encountered it? Specifically, your sample space $S$ is your set of $6$ outcomes, your event space $\Sigma$ is the power set of your sample space and your measure $\mathbb{P}$ is given by assigning $\frac{1}{6}$ to each singleton.
So your $S = \{1,2,3,4,5,6\}$ where each number represents the number you get. 
$\Sigma = \{A:A\subseteq S\}=\{\emptyset, \{1\}, \{2\}, ..., \{1,2\},...,\{1,2,3,4,5,\},..., S\}$. Basically the set of all subsets of S. Generally the event space is a $\sigma$-algebra, a set of sets of your sample space that is closed under complements, countable unions and has the empty set in it.
$\mathbb{P}$ is a function that is countably additive and assigns to each element in your event space (so each subset of $S$ here) a probability between $1$ and $0$ inclusive. Here we'll assign each singleton the probability $\frac{1}{6}$, the empty set the probability $0$, and any set that is the countable disjoint union of other sets in your event space will be given the sum of the probabilities of the sets in the union. We can then figure out what the probabilities of all the other sets in $\Sigma$ are just from this using properties of probability measures. e.g. $$\mathbb{P}(\{1\}\cap\{2\})=\mathbb{P}(\{1\})+\mathbb{P}(\{2\}) -  \mathbb{P}(\{1\}\cup\{2\}) = \frac{1}{6}+\frac{1}{6}-\frac{2}{6} =0$$
and the event in your question, of getting a $3$ has, by the definition of our probability measure, a probability of $\frac{1}{6}$.
The triplet $(S,\Sigma,\mathbb{P})$ is your probability space.
For your second question, I'm not really sure what you're asking, but all Hilbert spaces are Banach spaces, so if something is a property of all Banach spaces, it is a property of all Hilbert spaces. The converse is not true. 
