lutely certain event). The sum of all probabilities, which represent the combined likelihood of all possible events in X, is equal to unity. With the lotto, we know from experience that the odds of winning are quite small, and the odds of losing are, therefore, substantially high. For instance, we may assume: p(W)=0.000001, (3.1) p(L)=1-p(W)=0.999999, (3.2) meaning that the odds of winning are 1/1 000 000 or 1 in a million. How much information do I get when hearing the lotto result? Obviously, my expec- tations of winning being not very high, I won't learn much upon realizing that I have, indeed, lost. But in the contrary case, if I win, this is quite a phenomenal surprise! There Measuring information is a lot of information in the winning event, to the same extent there is little information in the losing event. Now we are looking for a formal way of defining the amount of information, which we could use as an objective measure reference I(x) for each possible event X. We may postulate that I(x) should approach zero for events close to absolute certainty (p(x) 1), and infinity for events reaching impossibility (p(x) 0). We must also require that for two events X1, x2 such that p(x1) < p(x2), we have I(x1) > I(x2), meaning that the less likely event is associated with greater information. An information measure I(x) that satisfies all these properties is the following: (tr)=108(a) (3.3) In the above, the function log represents the natural logarithm, with the unit of I(x) being called a nat. Note the minus sign in the definition's right-hand side, which ensures that the measure of information I(x) is a positive number. This definition only applies to events of nonzero probability (p(x) > 0), meaning that the definition does not apply to impossible events (p(x) = 0), or only in the sense of a limit. In our lotto example, we have I(W) = -log(0.000001) = 13.8 nats, (3.4) I(L) = -log(0.999999) = 0.000001 nat. (3.5) Thus, to win this lotto game represents some 14 nats of information, while losing means negligible inf