 to describe any of the die-roll outcomes with the same symbol length, i.e., in binary representation: X = 2 x = 3 - x110 x001, x011, Entropy Table 4.1 Calculation of entropy associated with the result of rolling two dice. The columns show the different possibilities of obtaining values from 2 to 12, the corresponding probability p and the product -plog2p whose summation (bottom) is the source entropy H, which is equal here to 3.274 bit/ symbol. Sum of dice numbers Probability (p) -plog2(p) 2=1+ 1 0.027 777 778 0.143 609 03 3=1+2=2+1 0.055 555 556 0.231 662 5 4=2+2=3+1=1+ 0.083 333 333 0.298 746 88 S=4+1=1+4=3+2=2+ 3 0.111 111 111 0.352 213 89 6 =5+1=1+5=4+2=2+4=33 0.138 888 889 0.395 555 13 =6+1=1+6=5+2=2+5=4+3 0.166 666 667 0.430 827 € 08 8=6+2=2+6=5+3=35=4 4 0.138 888 889 0.395 555 13 9=6+3=3+6=5+4=4+ 0.111 111 111 0.352 213 8 89 10=6+4=4+6=5+5 0.083 333 333 0.298 746 88 111=6+5=56 0.055 555 556 0.231 662 5 0.027 777 778 0.143 609 03 Source entropy = 3.274. where the first bit x is zero for all six outcomes. Nothing obliges us to attribute to each outcome its corresponding binary value. We might as well adopt any arbitrary 3-bit mapping such as: x = 1 x = 4 x=6 011. The above example illustrates a case where entropy and symbol information are equal, owing to the equiprobability property. The following examples illustrate the more general case, highlighting the difference between entropy and information. Two-dice roll The game consists in adding the spots obtained from rolling two dice. The minimum result is x = 2 (= 1 + 1) and the maximum is x = 12 (= 6 + 6), corresponding to the event space X = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The probability distribution p(x) was described in Chapter 1, see Figs. 1.1 and 1.2. Table 4.1 details the 11 differ- ent event possibilities and their respective probabilities p(x). The table illustrates that there exist 36 equiprobable dice-arrangement outcomes, giving first p(1) = p(12) = 1/36 = 0.027. The probability increases for all other arrangements up to a maximum correspo