Flipping Coins, Expected Value Interpretation

Say we have the following problem:

We are flipping some coin. The probability of it landing on heads is $$p$$. Is the expected number of coin flips we need to get the expected number of heads to be 1 different from the expected number of coin flips we need to get one head?

P.S. when I say are they the same I mean are they equivalent questions, not are the values of both the same (which they are).

I would think these are the same but I don't know how to describe why.

Let $$X\sim Ber(p)$$, where $$P(X=1)=p$$, and $$P(X=0)=1-p$$.
$$E(E(X=1))=E(X=1).$$
This is true because $$E(X=1)$$ is a constant (in fact, $$p$$), so you can take it out of the $$E(\cdot)$$ on the left hand side.
• To try and clarify my question, if I was to solve for the value on the right by taking indicator random variables $X_1,X_2...$ such that the probability they are $1$ is $p$, and then solving $np = 1$ for $n$ would I be incorrect in my methodology? – justaguy Feb 23 at 1:57