1. Certain events are not surprising: If some event $E$ will most definitely happen, $p(E)=1$, then that event occurring should provide us with no surprise (or new information): $H(E)=0$. 2. Impossible events are infinitely surprising: If some event $E$, has zero chance of occurring, $p(E)=0$, then we should be infinitely surprised that the event is occurring $H(E)=\infty$. 3. Non-Increasing: $I(p)$ should be non-increasing in p (i.e. the less likely event $E$ is, the more information one gains from it happening). 4. Continuity: $I(p)$ should be continuous in p. Intuitively, one would expect that a small change in p corresponds to a small change in the amount of information about $E$. 5. Continuity of Independence: If $E_1$ and $E_2$ are independent with probabilities $p_1>0$ and $p_2>0$, respectively, then $$I(E_1\cap E_2)=I(p_1*p_2)=I(p_1)+I(p_2)$$ This property is “reasonable” as $E_1$ and $E_2$ are independent.

$$\begin{array} {|r|r|}\hline b & \text{units of }I(p) \\ \hline 2 & \text{bits} \\ \hline e & \text{nats} \\ \hline 3 & \text{ternary units} \\ \hline q & \text{q-ary digits} \\ \hline \end{array}$$