e second lesson is that block codes also make it possible to increase the coding efficiency when the events in the original source exhibit a fair amount of mutual correlation, as we have seen with English digrams. However, we should not conclude that block codes are more efficient with such sources, and are less efficient with sources of mutually uncorrelated events. The last example shall prove this point. 18 See, for instance, www.ics.uci.edu/~dan/pubs/DC-Sec4.html. Optimal coding and compression Example 9.3: Two-event source; the roulette game We analyze the roulette game, as inspired by previous work, 19 but which is further developed here. This game originated in France in the seventeenth century, and was later imported to America with minor modifications, hence, the alternative expressions of French wheel or American wheel. To recall, the principle of the roulette is to make various types of bets on the ball landing at random in one of 36 spots, numbered from 1 to 36, of the rotating wheel. If the ball stops on the right spot number, the gain is 36 times the amount that was bet. To increase the odds on winning anything, bets can also be made on one, two, or four numbers at once, but this reduces the gain in the same proportion. To increase the variety of betting possibilities further, the wheel spots are also divided into families: (a) numbers 1-18 and 18-36, (b) odd and even numbers, (c) red or black, and (d) numbers 1-12, 13-24, and 25-36.4 A key feature of the roulette, which is not friendly to the players, is the existence of an extra green number, called 0 in the French wheel. The American wheel also has this number 0 plus a second one, called 00. When the ball lands on either 0 or 00, all bet proceeds go directly to the "bank." It is easy to establish that the odds on winning in the roulette game are relatively low, no matter how hard one may try with any playing strategy.21 Here, we shall simplify the game by assuming that the gambling exclusively concerns a single number selected from 1 to 36. The c