What are the configurations of a black hole? If black holes have entropy, they must have configurations. Since the entropy is calculated with a sum over configurations and their probabilities, $$S = \sum_i -p_i \log(p_i).$$ 


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*What are the configurations? 

*How do you measure the configuration of a black hole? 
Note: None of the responses have yet given me the answer that I want. I want to know what sort of measuring device could measure the configurations of a black hole? Let me give an example. Suppose that I wanted to understand a heat engine, like a car engine. Someone could explain to me thermodynamics and the physics of explosions, etc... etc... But what I want to know is that thermometers exist and that its reading change in a particular way if I put it next to one of the valves. A barometer is another such device that helps me understand an engine. What is the equivalent of a thermometer or barometer or whatever that could measure the configuration of a black hole? 
 A: There has been progress made to understand the Bekenstein-Hawking entropy from a microscopical point of view, in order to be able to identify the dynamics of black holes with our present understanding of thermodynamics.
The first step towards this was as mentioned in an answer is by Strominger and Vafa who considered type $\mathrm{II}$ string theory compactified on $K3\times S^1$, wherein a black hole solution can carry a charge $Q_F$ or $Q_H$ with respect to the $F$ and $\tilde H$ field-strengths respectively. It was found such black holes have Bekenstein-Hawking entropy,
$$S = \pi \sqrt{2Q_H Q^2_F}.$$
Their idea was to count BPS states, which for a super-symmetric sigma model, is equivalent to counting states which preserve a quarter of the super-symmetry, which are the $RR$ states in the right-moving vacuum.
The generating function for the degeneracies is bounded by the elliptic genus of the target space of the sigma model, and one finds an asymptotic result,
$$S \sim 2\pi\sqrt{Q_H\left(\frac12 Q^2_F+1\right)}$$
for $Q_H \gg 1$ which is in agreement with the Bekenstein-Hawking entropy for large $Q^2_F$. In other words, there is a relation between the entropy of the black holes in this theory, with the degeneracy of the BPS states.
A: Actually, the standard view is simpler and both much more intuitive as well as corresponding to black hole thermodynamic entropy. Theorists are trying different models to get the equivalent Shannon entropy. 
Consider the black hole horizon as being made up of a number N of Planck area patches. Each of those Planck area patches is the smallest area that can be considered, and roughly consider that it can be described by one bit - Ie, the $p_i$ of each Planck patch is 1/2. So it's Shannon entropy is, from your sum that defines the Shannon entropy, for one Planck area, 
$$S_i = -p_i log(p_i) = -1/2 log(2^{-1}) = -1/2(-1) = 1/2$$
and then simply S = $\Sigma S_i$ = N/2
N is the horizon area divided by the Planck area, $N = A_H/L_p$, 
so we get the equation for the BH entropy as Shannon's entropy times k, which is then 
$S_{BH} = kA_H/2L_p$
For some reason that I can't track it's off by a factor of 2, and the real answer from the Bekenstein-Hawking findings for BH entropy has a 4, so it's really
$S_{BH} = kA_H/4L_p$
The factor of 2 is not too worrisome, the argument used is not rigorous and it is not strange that it is off by a factor of2. But importantly it tells you what people think is the different configuration states of the BHs, basically in terms of what's there at each Planck sized patch of spacetime. 
The most rigorous derivation of the thermodynamic entropy was through the arguments and calculations of Bekenstein and Hawking where they got a rigorous expression for the equivalent temperaTure by considering how much the BH radiates which Hawking calculated in terms of quantum fields. The Shannon equivalent then depends on what you postulate, or can prove, what is the possibility for each Planck area. It is still not a rigorous answer, but you get it for instance from some string theories. If we ever get to an accepted quantum gravity theory it is expected/hoped it'll get the Bekenstein Hawking formula, at least in some limit. Clearly it is an area of research.  See a treatment of the BH entropy in http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy
and a paper at https://arxiv.org/pdf/1008.0946.pdf
There are other good papers on it you can google. 
You'd have to measure the number of elementary patches in the BH horizon to try to measure the entropy, not easy to do at a distance, or for that matter anywhere. Or the temperature of the blackbody radiation if you can get convinced that it is the same as the Shannon entropy times k. 
