Determining average number of hash calls to get a hash that begins with X I have been attempting to find an answer to a self-study question: 


  If I have a hash algorithm that outputs 256 binary bits, on average, how many times would I have to call the hash function so that the output begins with 15 zeros?


I am absolutely lost, but bear with me here - each bit has two possible values and assuming an uniformly random output, there are $2^{256}$ (or 1.1579209e+77) possible outputs from the hash function.

So approximately, how many calls to the hash function ($H$) would be required to get an ouput value that begins with those 15 zeros?  Note that I am not looking for the probability of finding such a value (although that could be useful to know as well).
Assuming the output is uniformly random, the probability of any given bit being 0 is $\frac{1}{2}$. So the probability that $b_0,b_1,b_2...b_{14} = 0$ is ${(\frac{1}{2}})^{15}$.

On average, it will require $2^{15}$ calls to find a hash where the first 15 bits are 0.