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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.

Thanks in advance :)

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Let $X\sim Ber(p)$, where $P(X=1)=p$, and $P(X=0)=1-p$.

You are asking whether

$$E(E(X=1))=E(X=1).$$

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.

What do you mean when you say "are those equivalent questions"?

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  • $\begingroup$ 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? $\endgroup$ – justaguy Feb 23 at 1:57

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