TITLE: Why do lower probability messages contain more information?
QUESTION [8 upvotes]: This question comes from here.
 Suppose messages $m_1, m_2, \ldots$ can be sent (through a channel) to a receiver with probabilities $p_1, p_2, \ldots$. The amount of information transferred when a message $m_k$ is successfully received is defined as
$$I_k = \log \frac{1}{p_k}.$$
The authors of the document I linked make the following claims about $I_k$:

It is intuitive: the occurrence of a highly probable event carries little
information ($I_k = 0$ for $p_k = 1$).
We gain more information when a less probable message is received
($I_k > I_l$ for $p_k < p_l$).

So my question is:
Why should a more probable message carry less information? Is it simply because successfully transmission of a longer message should have lower probability and a longer message can contain more meaning? E.g. if $m_1$ is the message "You're gonna die", and $m_2$ is the message "You're gonna die tomorrow", then successfully transmitting $m_2$ should be at least as hard (i.e. at least as improbable) as transmitting $m_1$ and $m_2$ clearly has more information.

REPLY [5 votes]: You are at "Sahara desert". Then you meet someone and the guy says:
- Today it did not rain!
How much information did you get? A few? A lot? Why?