The relationship between difficulty of guessing/predicting an unknown random variable $X$ and entropy (or entropies) is quite a delicate one, and depends on the assumptions about the attack scenario. In particular, the direct use of Shannon entropy can give misleading results.
It was shown by John Pliam (see preprint ) that if $X$ is a discrete random variable with $M$ points in its support $H(X)$ can be close to its maximum value $\log M$ while the probability of an optimal guessor discovering the actual value of $X$ in $k$ sequential questions is much less than $2^{H(X)}=M.$
Moreover, not only the expected number of guesses, but arbitary moments
of the number of guesses can be related to Renyi entropies of various order. Renyi entropies are a family of entropies $H_{\alpha}(X)$ with parameter $\alpha \in [0,1)\cup(1,\infty)$ which satisfy $$\lim_{\alpha \rightarrow 1}H_{\alpha}(X)=H(X),$$where $H(X)$ is the Shannon entropy of $X.$
One result on the expectation is that the expected number of guesses to
determine
a random variable $X$ (for the optimal guessing sequence) is
upperbounded by
$$2^{H_{1/2}(X)-1}$$
where $H_{1/2}(X)$ denotes the Renyi entropy of order $\alpha=1/2$ and is also called guessing entropy. Moreover, it is lowerbounded by (for all guessing sequences)
$$\frac{2^{H_{1/2}(X)}}{1 + \log M } \approx 2^{H_{1/2}(X)} / H_{max}(X)$$
where $H_{max}(X)$ is the maximum entropy $\log M$ in either the Renyi
or the Shannon case. Papers by J L Massey, Arikan, Boztas and others have followed this train of thought.