What is the probability of success given two trials with a 1/16 chance each? I have never been able to wrap my head around probability, and I often find that my intuition is wrong.  In this case, I don't even have intuition as to where to begin.
If I have two trials, each with a 1/16 chance of success, what are the chances that either or both of them result in success?  How, mathematically, do you arrive at the correct probability?  How, intuitively, can I understand this number?

It also might be helpful to think of this in terms of geometric distribution.
Geometric distribution represents the number of failures before you get a success and is defined as:
$(1-p)^{n-1} \times p$
In your case, what we have is two geometric distributions added together:
$\left(\frac{15}{16}^{0} \times \frac{1}{16}\right)+\left(\frac{15}{16}^{1} \times \frac{1}{16}\right) = \frac{31}{256}$
The reason we add it together is the first term assumes that you had no failure and your first trial was a success, your second term assumes that you had one failure and your second trial was a success.
