nding to the event X = 7 with p(7) = 6/36 = 0.166. Summing up the values of - p(x) log p(x), the source entropy is found to be H = 3.274 bit/symbol. This result shows that, on average, the event can be described through a number of bits between 3 and 4. This was expected since the source has 11 elements, which requires a maximum of 24 = 4 bits, while most of the events (namely, X = 2, 3, 4, 5, 6, 7, 8, 9) can be coded 4.2 Entropy in dice in principle with only 23 = 3 bits. The issue of finding the best code to attribute a symbol of minimal length to each of the events will be addressed later. The 421 The dice game called 421 for short was popular in last century's French cafés. The game uses three dice and the winning roll is where the numbers 4, 2, and 1 show up, regardless of order. The interest of this example is to illustrate Shannon's property (3), which is described in the previous section and also in Appendix B. The probability of obtaining x = 4, 2, 1, like any specific combination where the three numbers are different, is p(421) = (1/6)(1/6)(1/6) X 3! = 1/36 = 0.0277 (each die face has 1/6 chance and there are 3! possible dice permutations). The probability of winning is, thus, close to 3%, which (interestingly enough) is strictly equal to that of the double six winner (p(66) = (1/6)(1/6) = 1/36) in many other games using dice. The odds on missing a 4, 2, 1 roll are p(other) = 1 - p(421) = 35/36 = 0.972. A straightforward calculation of the source entropy gives: H(421, other) = -p(421) p(421) p(421)] log2[1 - p(421)] (4.9) The result shows that for certain sources, the average information is not only a real number involving "fractions" of bits (as we have seen), but also a number that can be substantially smaller than a single bit! This intriguing feature will be clarified in a following chapter describing coding and coding optimality. Next, I shall illustrate Shannon's property (3) based on this example. What we did consisted in partitioning all possible events (result of dice rolls) into two subcate- g