Donald’s-Nathanson’s System
I always make bets on Red. Let us assume that the initial bet is 1$. After Black has occurred, I raise the bet by one, and after the Red has occurred, I lower it by one. However, what should I do if I have bet a dollar on Red and have won? According to T. Donald’s system, the bet must be unchangeable, as there are neither zero nor negative bets in the game. “Why not?” the mathematician thought, and he tried. It appeared to be quite interesting.
So that not to abandon the canons of the initial system, after you have made a bet on Red and have won, you should lower the bet by one. If you bet 1$, the next bet must be equal to zero. It is clear what a zero bet is, isn’t it? You just miss the next turn of the roulette. By the way, bet zero namely on Red and watch carefully what is going to occur, in order to know what bets you are going to make next time. Let us suppose that Red has occurred again. You have won and must lower the bet once more. The next bet (according to the system) must be equal to minus one.
In addition, what is a negative bet on Red? This is a bet on Black! Whatever may happen further on, the rule is the same: when Black occurs, the bet rises; when Red does, the bet falls.
For example, the wheel makes the first three turns and every time Red occurs. After the first turn of the roulette, we win 1$. Next time we bet 0, and in the third time minus 1$ (a dollar on Black).
Before the wheel makes the fourth turn, we must reduce the bet to minus 2$. We bet 2$ on Black.
It can be proved that if Red and Black occur N times out of N turns of the roulette, the winning will make precise N of the initial bets. There exists the property of the invariance; regardless of how often Red (or Black) occurs: the succession in which Red replaces Black does not influence the size of the winning.
Let us suppose that the roulette has been turned for 36 times. For instance, if Red has occurred for 20 times, the winning will make 14$ while the initial bet is 1$. If Red has occurred only for 17 times, you will also win 14$. T. Donald reckoned on the nearness of this frequency. Nathanson only followed him and intensified the system.
Let us remember Zero in order to accomplish the pattern. According to T. Donald, when zero occurs, one should raise the next bet. In Nathanson’s modification, one must raise it in accordance with the module. In other words, if the bet is positive, one should raise it by one, if it is negative, one should lower it. Unfortunately, the appearance of zero affects the fine property of the invariance, and it will be impossible to find out your profit. It is suffice if zero occurs only once out of 36 turns of the roulette.
Let us assume that the bet is positive when zero occurs. In this case, zero is equivalent to Black. Thus, the profit is determined according to that particular table. For example, when Red occurs for 20 times, Black – for 15 times, and zero – for 1 time, the winning will make 14$. However, don’t think that zero influences nothing. It decreases the probable frequency with which Red occurs.
Zero may also occur when the bet is negative. Now it is equivalent to Red. If Red has occurred for 20 times, the number of its occurrence is 21 due to zero. Instead of 14$ (according to the table) we actually win only 6$. On the other hand, if Red has occurred less than 18 times, your profit gets larger.
Finally, zero may occur when the bet is zero. You may act in any way: if you raise the bet, zero will be equivalent to Black, if you lower it – to Red. Nevertheless, look back to the prehistory of the game: if Red occurred more often, you should raise the bet; if it occurred rarer, just the other way round. Thus, you draw near to each other the frequencies with which both colors occur.
