Why do we use log2(1/p(xi)) in Shannon's entropy definition?

[dervast]

Hi to everyone i was reading today the wikipedia's article about information entropy
I need some help to understand why in the
http://upload.wikimedia.org/math/6/a/3/6a33010c16b1d526bc5daee924e3d363.png
entropy of an event we use the log2(1/p(xi))
I have read to the article that
"An intuitive understanding of information entropy relates to the amount of uncertainty about an event associated with a given probability distribution. "
Then why only the first part of the equation Sump(xi) is not enough. p(xi) denotes the amount of uncertainty of an event..

 

[fresh_42]

The entropy is an expectation value, namely $H(X)=E(X)=-\sum_{s\in S} p(X=s) \cdot \log_2(p(X=s))$; the weighted sum of all informations. The information of an event $s$ over a binary alphabet is defined as $\log_2 \left( \frac{1}{p(X=s)} \right)$. The rest follows from this.

There are other possible measures of information, but the one above is which Shannon defined and considered.

The logarithm is basically a length, and $\frac{1}{p(X=s)}$ the statistical relevanz: An event which occurs for sure doesn't carry any information. But the less probable an event is, the more information is carried. Imagine an encoded three letter word. An 'x' in the middle carries more information about this word than an 'e' does. There are only a few words like 'axe', but many like 'bet', 'get', 'jet', 'let', 'set', 'men', etc.