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Mathematical Systems - Part 3

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In Mathematical Systems, Parts 1 and 2, we talked about the Martingale and Grand Martingale systems along with the "d'Alembert" (or the Pyramid System) and the "Labouchere," (also called the Cancellation System) in great detail.

In this installment of Mathematical Systems, we will look at the Fibonacci sequence and Oscar's Grind. These two systems will not wipe you out as quickly as the previous ones discussed, but as you now know, ALL mathematically based systems will cause you to lose your money eventually. This is how the house has been able to stay ahead of the long-term players and run a thriving business over the last couple of hundred years! Please remember that a system that dictates the size of your next wager, based on whether you won or lost the last bet, will wipe out your bankroll over an extended period!

The Fibonacci

Leonardo Pisan, better known as Fibonacci, was born in Pisa (now part of Italy) in 1170 A.D. Fibonacci was a member of the Bonacci family and traveled all around the Mediterranean as a boy with his father who held a diplomatic post. His keen interest in mathematics and his exposure to other cultures allowed Fibonacci to excel in solving a wide variety of mathematical problems. Fibonacci is probably best known for discovering the Fibonacci sequence, a sequence of numbers that readily exists in nature. Although, technically not a mathematical system per se, the sequence is often used in a losing or negative progression. The Fibonacci series is as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

The next number in the series is simply the sum of the previous two numbers. The starting number is 1. The second number calculated from 0 +1 (no number in front of the first 1) and is 1 again. The next number is 1 +1 or 2, then 1 +2 for 3, then 2 +3 = 5 and 5 +3 = 8, etc. The system works similarly to the Labouchere or cancellation system, only the player starts out with an empty line. If the first bet is won, then the sequence is over and the player has won. No numbers need to be written down. If the first bet is lost, then a line is started and a "1" is written down. The next number in the sequence represents the following wager size. If this bet is lost, then it is added to the end of the line. As each bet is lost, it is added to the end of the series. If a bet is won, the last number in the series is crossed out. An example here will help clarify things:

1.) Bet 1 unit and lose: 1 –1 units
2.) Bet 1 unit and lose: 1-1 –2 units
3.) Bet 2 units and lose: 1-1-2 –4 units
4.) Bet 3 units and win: 1-x-x –1 units
5.) Bet 1 unit and lose: 1-1 –2 units
6.) Bet 2 units and lose: 1-1-2 –4 units
7.) Bet 3 units and lose: 1-1-2-3 –7 units
8.) Bet 5 units and win: 1-1-x-x –2 units
9.) Bet 2 units and lose: 1-1-2 –4 units
10.) Bet 3 units and win: 1-x-x –1 units
11.) Bet 1 unit and lose: 1-1 –2 units
12.) Bet 2 units and win: x-x +0 units
13.) Bet 1 unit and win: stop +1 unit
-Series has been won-

Our player starts with a one unit loss, so a "1" is recorded to start the line. Another "1" is added after the second wager of one unit loses. The third stake requires a two-unit wager and loses, so a "2" is added. The fourth bet of three units finally wins and the "1-2" can be cancelled from the line. Because each wager adds up to the previous two bets, the last two numbers on the line can be crossed out when a bet wins. The next three bets lose, escalating our eighth stake up to five units. Our player experiences a win at this level, allowing him to cancel out the "2-3" at the end of the line. The ninth bet of two units loses, so the line grows to "1-1-2." A win, loss and win on the tenth, eleventh and twelve wagers finally wipe out the betting line. The player needs and gets a win at this point to go up a net profit of one unit and win the sequence.

With only five wins and eight losses, this particular sequence of wins and losses is tough, but our player is able to pull it out. On the eighth wager, the stake reaches a high of five units. If that bet had lost, our player would be twelve units in the hole. At a $5 unit size, that equates to a $60 deficit. The next wager from here would be eight units and another loss would put him back 20 units total. If you elect to use the Fibonacci, I would highly recommend that you limit your top bet to five units. If you lose your wager at this level, then abandon the series. Things get ugly too quickly from here. Stop and regroup. Let's take the Fibonacci up to twelve straight losses to see how quickly the wagers can mount:

1.) Bet 1 unit and lose: 1 –1 units
2.) Bet 1 unit and lose: 1-1 –2 units
3.) Bet 2 units and lose: 1-1-2 –4 units
4.) Bet 3 units and lose: 1-1-2-3 –7 units
5.) Bet 5 units and lose: 1-1-2-3-5 –12 units
6.) Bet 8 units and lose: 1-1-2-3-5-8 –20 units
7.) Bet 13 units and lose: 1-1-2-3-5-8-13 –33 units
8.) Bet 21 units and lose: 1-1-2-3-5-8-13-21 –54 units
9.) Bet 34 units and lose: 1-1-2-3-5-8-13-21-34 –88 units
10.) Bet 55 units and lose: 1-1-2-3-5-8-13-21-34-55 –143 units
11.) Bet 89 units and lose: 1-1-2-3-5-8-13-21-34-55-89 –232 units
12.) Bet 144 units and lose: 1-1-2-3-5-8-13-21-34-55-89-144 –376 units

This last example demonstrates how the bets can mount in a losing string of twelve losses. The chances of losing twelve straight on a double zero roulette wheel are (20/38)^12 = 0.0004518, or about 1 shot in 2213. The purpose here was to show a range of cumulative losses and let the system player decide where to draw the line. Some authors show the Fibonacci sequence and omit the first "1" in the series. That's fine, but the shortened version is a little more aggressive than the "full" Fibonacci. You will lose a bit more money on average with this abbreviated variation. Overall, the Fibonacci sequence does not fare to badly. This system can be fun and not too damaging if you limit your top bet to five units.

Oscar's Grind

The first reference I can find regarding this more modern betting system appeared in Allan Wilson's "The Casino gambler's Guide," copyright 1965. Wilson was intrigued with this system after a dice player named "Oscar" produced detailed records showing modest, but consistent profits. Wilson ran 280,000 sequence simulations on an IBM 790 mainframe computer that was available to him. The analysis showed that while Oscar was a bit on the luckier side, his claims were at least possible. Now remember that Oscar was a pass line bettor only attempting to buck a –1.414% house edge as compared to a –5.263% house edge for double zero roulette. In addition, Oscar had a mega-bankroll and the willingness to risk it all for a one unit per cycle win.

Let's look at the details of the "Grind." The system has the player bet one unit. If he wins, the sequence is over and a new one can be initiated. If the wager is lost, then the next bet will be the same size as the one just lost. Whenever a bet is won, the next stake is one unit larger, unless it causes the bettor to net more than one unit of profit for the sequence. At that point, just enough is wagered to net one unit if the bet wins. That's it! A sample sequence might look like this:

1) Bet 1 unit and lose: -1 unit
2) Bet 1 unit and win: +0 units
3) Bet 1 unit and lose: -1 unit
4) Bet 1 unit and lose: -2 units
5) Bet 1 unit and lose: -3 units
6) Bet 1 unit and win: -2 units
7) Bet 2 units and win: +0 units
8) Bet 1 unit and lose: -1 unit
9) Bet 1 unit and win: +0 units
10) Bet 1 unit and win: +1 unit
-Series has been won-

The player starts with a loss so his second stake remains at one unit. This bet is won, putting him back to even. Because he is only seeking a one-unit win for the progression, he does not escalate his bet to two units. Bets 3 through 5 are losses so he stays with a one-unit stake. After the sixth bet wins, he now increases his wager to two units. The seventh bet also wins, but again he only needs a one unit bet to win the sequence. The eighth bet loses so the ninth wager is one unit. Finally, the tenth bet wins and our player wins the entire progression. Notice that out of ten total wagers, nine were only one unit in size. This system tends to be more conservative and less volatile. The sequence illustrated above contained five wins and five losses. I like the fact that this system does not quickly escalate your losing wagers and blindside you like some of the others. However, as your losses outnumber your wins, the amount you must wager after a win will steadily mount. Sometimes in probability, a tree diagram is used to show all of the possible outcomes one can encounter. The rounded boxes are terminal events, ending the betting progression. This is either done by winning the progression (four ways to win) or reaching the fifth bet, where we have elected to stop the progression to keep things manageable. (Taking the negative progression further, will actually lose you more money). Each terminal event contains a percent chance that you will end the progression with that outcome. Look at the following tree diagram to see all of the possibilities for a progression up to five bets deep:

Oscar's Grind Tree Diagram

(5-bet progression using a $5unit)

Total probabilities of sequence endning events = 1.00 or 100%

You will win the $5 progression a larger portion of the time (68.34%). However, the smaller portion of the time your losses will be much larger and overall you will come out on the negative side of the ledger. Some calculations will help clarify this:

1) (18/38) or 47.37% of the time you will win $5 after the first bet.
2) (20/38) x (18/38)^2 or 11.81% of the time you will win $5 after the third wager.
3) (20/38)^2 x (18/38)^2 or 6.22% of the time you will win $5 after the fourth bet.
4) (20/38)^2 x (18/38)^3 or 2.94% of the time you will win $5 after the fifth bet.

The rest of the outcomes are a result of cutting the progression off after the fifth wager:

5) (20/38)^3 x (18/38)^2 or 3.27% of the time you will break even, $0
6) 2 ways, 2 x [(20/38)^3 x (18/38)^2] or 2 x 3.27% of the time you will lose $5
7) (20/38)^3 x (18/38)^2 or 3.27% of the time you will lose $10
8) 2 ways, 2 x [(20/38)^4 x (18/38)] or 2 x 3.63% of the time you will lose $15
9) 2 ways, 2 x [(20/38)^4 x (18/38)] or 2 x 3.63% of the time you will lose $20
10) (20/38)^5 or 4.04% of the time you will lose all five bets, or $25

A weighted summation of the outcomes gives us the following:

68.34% of the time we win $5;

(0.6834) x $5 win

= +$3.42

3.27% of the time we break even;

(0.0327) x $0 win

= +$0.00

2 x 3.27%, or 6.54% we lose $5;

(0.0654) x $5 loss

= –$0.33

3.27% of the time we lose $10;

(0.0327) x $10 loss

= –$0.33

2 x 3.63%, or 7.26% we lose $15;

(0.0726) x $15 loss

= –$1.09

2 x 3.63%, or 7.26% we lose $20;

(0.0726) x $20 loss

= –$1.45

4.04% of the time we lose $25;

(0.0404) x $25 loss

= –$1.01

Total net profit (+), or loss (–)

= –$0.79

As you can see, the net result for a $5 starting unit, up to five bets deep, averages a loss of 79 cents per progression. Another way to look at it is for every 100 times that you run the progression, you can expect to be down about $79.00 and this is one of the more conservative betting systems! A larger unit size or deeper progression will lose even more money. Like all other mathematical systems, this staking system will lose over a sustained period. They all have their Achilles heel. The "Grind" seems to stall in choppy games. You increase your stake after a win and then lose the larger wager, so you lower your next bet but you win that one. This back-and-forth-type sequence can add losses up quickly. Where Oscar's Grind will excel is in streaky games. The Grind minimizes your betting level if you are amidst a string of losses. It also directs you to gradually increase your wagers during a streak of wins, helping to optimize profits. This can be one of the safer systems to use if you limit your maximum bet size or impose a stop-loss parameter for choppy games. If you plan to use Oscar's Grind, I would recommend a stop-loss of about ten, no more than 12 units per cycle.

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