Omitting a preposition to avoid repetition

What is correct/better?

1. We consider questions related to analysis of real numbers and to probability theory.

2. We consider questions related to analysis of real numbers and probability theory.

An additional question: I do not need "the" before the word "analysis" in those examples, do I?

If you are talking to mathematicians, you don’t need the second “to”, because your audience knows that “analysis of real numbers” and “probability theory” are separate things, and they hear those as separate chunks. So, mathematicians hear the sentence as following this pattern:

Let’s go to dinner and a movie.

Let’s go to the Museum of Natural History and a movie.

Since you know math, I’ll use parentheses to show the grouping in the listener’s mind:

Let’s go to (the Museum of (Natural History)) and (a movie).)

However, if your listener had never heard of the Museum of Natural History (and couldn’t see the capitalization), your listener could hear it this way:

Let’s go to (the Museum of (Natural History and a Movie)).

In general, how objects get divided among prepositions depends on how familiar phrases, prior knowledge, and common sense influence how a listener “chunks” the sentence.

In your example, you can avoid even the small potential for ambiguity by using the common phrase “real analysis” so you don’t need to say “of”:

We consider questions related to real analysis and probability theory.

Conceivably, a naïve listener could hear that as:

We consider questions related to (real (analysis and probability theory)).

but the phrase “real analysis” is so familiar, anyone in the field will hear it as:

We consider questions related to ((real analysis) and (probability theory)).

You don’t need “the” before “analysis”, “real analysis”, or “probability theory”. Most names of fields of study do not get an article. For example, one studies “biology”, not “the biology”. One exception is “the calculus”, but while some textbooks have that as their title, I’ve never heard anyone say it in conversation. There are other exceptions, but that’s probably best addressed in a separate question.