how to interpret the uncertainty (information) of a single event (observation)?

Question

I have trouble understanding the following situation:
Let $X$ be a random variable which takes two values $x_1,x_2$ with probabilities $P(x_1)=1/4,P(x_2)=3/4$. Then the entropy or uncertainty of $X$ is:
$H(X)=0.811$.
However by definition the information of an observation $X=x_1$ is $h(x_1)=-log(P(x_1))=2$.
I interpret this as if we observe that $X=x_1$ then we get 2 bits of information (is it right?). But as Shannon states that information is the reduction of uncertainty, after observing $X=x_1$ we are now totally aware of the value of $X$, hence the uncertainty of $X$ reduced to $0$. And according to this reasoning we get $H(X)$ bits information. Apparently there is something wrong since the two values are not equal, but what is the problem?

And how does the entropy of $X$ link to the information $h(x_1)$ and $h(x_2)$?

Answer 1

if we observe that $X=x_1$ then we get 2 bits of information (is it right?)

Basically yes.

And when you observe that $X=x_2$ then you get $0.415$ bits of information. This happens with frequency $\frac 34$.

Then in average you get $\frac{1}{4} 2 + \frac{3}{4} 0.415 = 0.811$ bits of information.

Does this answer your doubt?