The relevance of Shannon entropy is to repeated sampling: Given $n$ independent samples from a source with binary Shannon Entropy $k$, you can extract $nk(1+o(1)$ i.i.d. uniform bits as $n$ tends to infinity with probability tending to 1. This follows e.g. from the Keane-Smorodinsky  finitary isomorphism theorem. See also - below.
 M. Keane and M. Smorodinsky (1979), Bernoulli schemes of the same
entropy are finitarily isomorphic. Annals of Math. 109, 397–406.
 P. Elias (1972), The efficient construction of an unbiased random sequence. Ann. Math. Statist. 43, 865–870.
 D. E. Knuth and A. C. Yao (1976), The complexity of nonuniform
random number generation. Algorithms and complexity (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1976), 357–428. Academic Press, New York.
 Y. Peres (1992), Iterating von Neumann’s procedure for extracting random bits. Ann. Stat. 20, 590–597.
 Harvey, Nate, Alexander E. Holroyd, Yuval Peres, and Dan Romik. "Universal finitary codes with exponential tails." Proceedings of the London Mathematical Society 94, no. 2 (2007): 475-496.