Editing Information content (section)

=== Monotonically decreasing function of probability ===
For a given [[probability space]], the measurement of rarer [[event (probability theory)|event]]s are intuitively more "surprising", and yield more information content, than more common values. Thus, self-information is a [[Monotonic function|strictly decreasing monotonic function]] of the probability, or sometimes called an "antitonic" function.

While standard probabilities are represented by real numbers in the interval <math>[0, 1]</math>, self-informations are represented by [[extended real number]]s in the interval <math>[0, \infty]</math>. In particular, we have the following, for any choice of logarithmic base:

* If a particular event has a 100% probability of occurring, then its self-information is <math>-\log(1) = 0</math>: its occurrence is "perfectly non-surprising" and yields no information.
* If a particular event has a 0% probability of occurring, then its self-information is <math>-\log(0) = \infty</math>: its occurrence is "infinitely surprising".

From this, we can get a few general properties:

* Intuitively, more information is gained from observing an unexpected event—it is "surprising". 
** For example, if there is a [[wikt:one in a million|one-in-a-million]] chance of Alice winning the [[lottery]], her friend Bob will gain significantly more information from learning that she [[Winning the lottery|won]] than that she lost on a given day. (See also ''[[Lottery mathematics]]''.)
* This establishes an implicit relationship between the self-information of a [[random variable]] and its [[variance]].