According to Wikipedia, the information entropy $H(X)$ of the random variable $X$, with possible outcomes $x_i$, each with probability $P_X(x_i)$, is given by the average information content (average self-information) of those outcomes:
$H(X) = \sum_i P_X(x_i) I_X(x_i)$
The information content $I(x)$ in turn is given (for some base $b$, normally 2) by:
$I(x) = -\log_b{P(x)}$
Now my question is: Can the entropy $H$ directly be interpreted as some kind of information content $I$? In other words, is there a formula for some $\varphi$ such that the following is true?
$H(X) = I(\varphi)$
Intuitively this would require the definition of some kind of probability $P(\varphi)$ which is high if the entropy is low (since $I(\varphi) = -\log_b{P(\varphi)}$).
If you have a random event $\Phi$ that can have one of $n$ possible outcomes with the same probability $P(\phi_i)=1/n$, then $$ H(\Phi) = \sum_{i=1}^n\frac1n I(\phi_i) = I(\phi). $$
Thus, if we forget that we can't have a non-integer number of outcomes, we can say that for a random variable $X$, we can define a new random variable $\phi$ that can have $n=b^{H(X)}$ outcomes with the same probability. And in that case $H(X)=I(\phi)$.
In other words, by definition, entropy is the average information content of all the outcomes. If all the outcomes have the same information content, then the entropy equals to the information content of any of the outcomes.