As in the previous case, the game with one die is uninteresting and we shall start by analyzing the case of two dice.
Recall that in the present game, we start with a specific ini-tial configuration, which for the next few games will always be the all-zeros configuration. Suppose you chose the sum = 0, arguing that since you know the result of the zeroth step, which is sum= 0, you are guaranteed to win the zeroth step. Indeed, you are right. What should I choose? Suppose I chose sum= 2, the largest possible sum in this game. Recall that in the previous game with two real dice, the minimum sum= 2, and the max-imum sum= 12, had the same probability1/36. Here, the rules of the game are different. If you choose sum= 0, and I choose sum= 2, you win with probability one and I win with probabil-ity zero on the zeroth step, so you do better on the zeroth step.
What about the first step? You will win on the first step if the die, chosen at random and tossed, has the outcome of zero. This occurs with probability1/2. What about me? I choose sum = 2.
There is no way to get sum = 2 on the first step. On the first step, there are only two possible sums, zero or one. Therefore, the probability of my winning on the first step is zero too. So you did better both on the zeroth and first steps.
What about the next step? It is easy to see that your chances of winning are better on the second step as well. In order for me to win, the sum must increase from zero to one on the first step, and from one to two on the second step. You have more paths to get to sum = 0 on the second step; the sum = 0 can be realized by, “stay at sum = 0” on the first step, and “stay
96 Entropy Demystified first step, and decrease to sum= 0 on the second step.
Figure (4.2) shows two runs of this game,3each run involving 100 steps. It is clear that after many steps, the number of “visits”
to sum = 0 and to sum = 2 will be about equal. Although we started with sum = 0, we say that after many steps, the game loses its “memory” of the initial steps. The net result is that you will do a little better in this game.
What if you choose sum = 0, and I choose sum = 1? In this case, you will win with certainty on the zeroth step. On the second step, you have a probability, 1/2, of winning, and I have a probability, 1/2, of winning. But as we observe from the “evolution” of the game, after many steps, the game will visit sum = 0 far less frequently than sum = 1. Therefore, it is clear that after many games, I will be the winner in spite of your guaranteed winning on the zeroth step.
3We refer to a “run,” as the entire game consisting a predetermined number of
“steps.”
Let us Play with Simplified Dice 97
Let us leave the game for a while and focus on the evolution of the game as shown in the two runs in Fig. (4.2). First, note that initially we always start with sum= 0, i.e., with configuration {0, 0}. We see that in one of the runs, the first step stays at sum= 0, and in the second run, the sum increases from zero to one. In the long run, we shall be visiting sum= 0 about 25% of the steps; sum= 2, about 25% of the steps; and sum = 1, about 50% of the steps. The reason is exactly the same as in the case of playing the two-dice game in the previous chapter. There is one specific configuration for sum = 0, one specific configuration for sum= 2, but two specific configurations for sum = 1. This is summarized in the table below.
Configuration {0, 0} {1, 0} {0, 1} {1, 1}
Weight 1 2 1
Probability 1/4 2/4 1/4
This is easily understood and easily checked by either exper-imenting with dice (or coins), or by simulating the game on a computer.
In Fig. (4.2), you can see and count the number of visits at each level. You see that the slight advantage of the choice of sum= 0 will dissipate in the long run.
Before we proceed to the next game with four dice N = 4, we note that nothing we have learned in this and the next game seems relevant to the Second Law. We presented it here, mainly to train ourselves in analyzing (non-mathematically) the evolu-tion of the game, and to prepare ourselves to see how and why new features appear when the number of dice becomes large.
These new features are not only relevant, but are also the very essence of the Second Law of Thermodynamics.
98 Entropy Demystified
If you run many games like this on a computer, you might encounter some “structures” in some of the runs, for instance, a sequence of 10 consecutive zeros, or a series of alternating zeros and ones, or whatever specific structure that you can imagine.
Each specific “structure”, i.e., a specific sequence of results is possible and if you run many games, they will occur sometimes.
For instance, the probability of observing the event sum= 0, in all the 100 steps is simply (1/2)100 or about one in 1030 steps.