Do you apply double Proficiency from Expertise on top of normal Proficiency?

Question

So I am playing D&D 5e with a lvl 1 Elven Rogue as my character. I'm confused about expertise for my Perception. My wisdom is 13, so my modifier is +1. Which makes my Perception 1.

But as an elf, you get a proficiency bonus (+2) on Perception. So now my perception is 3.

But with expertise, you are proficient, and you double that proficiency. So does that mean I am 3+2=5, or am I 3+4=7? I remember reading somewhere that you can't apply your proficiency bonus twice for one roll, but can you add a race and expertise bonus to the same roll?

Answer 1

Expertise doesn't add a bonus; it doubles the bonus that you're getting for being proficient. So rather than thinking of it as 1+2+2, think of it as 1+(2×2). Also note that as you gain levels your proficiency bonus will go up slowly, so at level 5, it will become 1+(3×2).

Expertise

At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with thieves’ tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies. (Basic Rules, p. 27)

Answer 2

Expertise doubles proficiency, since proficiency is proficiency (you don't get double proficiency if the same proficiency is available from your class and your background; etc) what happens is that you double your proficiency bonus and add your ability modifier. So 1+(2x2) = 5, per the comment above by Derek Stucki

In general, if in doubt, the answer is almost always the less powerful of the possibilities.