POSITIVE EXPECTATION ON ROULETTE CAN BE PROVED MATHEMATICALLY!

[Blue_Angel]

Dear readers,

this is a thread which requires your full attention in order to fully comprehend the hidden opportunity.

Let's start with the basic about the Law Of the Third:

In every 37 spins cycle there are 24 numbers which have appeared once or more and 13 sleepers
(numbers which haven't appeared within the last 37 spins)
Those are average numbers and deviations exist, for example I've seen up to 30 different numbers to show up within the 37 last spins and the least were 18 different numbers.
Those extremes are from my experience during gambling sessions and not from simulations.
In order to find the average we should determine the extremes,or in other words the limits, in that case 18 and 30 are the limits.

18 19 20 21 22 23 24 26 26 27 28 29 30 start by discarding the outer numbers from both sides:

First discarding 18 and 30
Second discard 19 and 29
Third discard 20 and 28
Fourth discard 21 and 27
Fifth discard 22 and 26
Sixth discard 23 and 25

And we conclude to 24 which is the average total of the appeared numbers within 37 last spins.
Another way to calculate the average is to add all the totals and then divide the total sum with the total of the averages,for example:

18 different numbers within the last 37 spins happened 1 time
19 different numbers within the last 37 spins happened 2 times
20 different numbers within the last 37 spins happened 3 times
21 different numbers within the last 37 spins happened 4 times
22 different numbers within the last 37 spins happened 5 times
23 different numbers within the last 37 spins happened 6 times
24 different numbers within the last 37 spins happened 7 times
25 different numbers within the last 37 spins happened 6 times
26 different numbers within the last 37 spins happened 5 times
27 different numbers within the last 37 spins happened 4 times
28 different numbers within the last 37 spins happened 3 times
29 different numbers within the last 37 spins happened 2 times
30 different numbers within the last 37 spins happened 1 times

[(1 x 18)+(2 x 19)+(3 x 20)+(4 x 21)+(5 x 22)+(6 x 23)+(7 x 24)+(6 x 25)+(5 x 26)+(4 x 27)+(3 x 28)+(2 x 29)+(1 x 30) / 49] =>

[(18+38+60+84+110+138+168+150+130+108+84+58+30) / 49]=>

1176 / 49 = 24

This means that if someone was betting one number for 37 spins, 24 times he/she would have won and 13 times he/she would have lost.
To calculate the total sum of lost bets is easy:
13 times multiplied by 37 (1 x 37) equals minus 481 units.

In order to find the total amount of won bets, we should again calculate the averages as we did above,let's see:
We start from the middle numbers within a 37 spin cycle and add 2 more numbers from both sides till we have a total of 24.
18th and 19th spins
17th and 20th spins
16th and 21st spins
15th and 22nd spins
14th and 23rd spins
13th and 24th spins
12th and 25th spins
11th and 26th spins
10th and 27th spins
9th and 28th spins
8th and 29th spins
7th and 30th spins
So we have 24 wins from different spins, now let's calculate the total amount of net profit.

29+28+27+26+25+24+23+22+21+20+19+18+17+16+15+14+13+12+11+10+9+8+7+6 = 420

But since we would lose 481 - 420 = -61 loss, therefore the definition negative expectation.

Do we agree so far?
Is everything clear?

Let's see how we could turn the negative expectation into positive without changing a thing in terms of probability, in fact with exactly the same results which have lead us to the negative balance above.

The average expectation is to get twice the wins for every loss, with my method you only need approximately 1 positive cycle for every negative cycle, which means that the proportion of 2 to 1 changed to 1 to 1.

Let me explain how this is possible, the only thing which I'm not going to reveal you here is the criteria which I'm using to select the betting number.
You could pick a random number, or your "lucky" number or anything else you like.

I start flat betting 1 number with 1 unit for 36 spins.
When the betting number appears, no matter in which spin, I re-bet the same number plus its neighbour regarding the wheel layout (right or left doesn't matter)

So now I'm flat betting 2 numbers with 1 unit each for the next 18 spins.
When one of the two betting numbers hits within eighteen spins, then I re-bet those two numbers and adding the other wheel neighbour, thus in total three numbers for the next 12 spins, always flat bet 1 unit each.

When one of the three betting numbers comes, I add one more neighbour,this time from the table layout.
So far we have 4 numbers to flat bet with 1 unit each for the next 9 spins.
Let's see if you keep on winning what happens:

5 numbers flat bet with 1 unit each for the next 7 spins
6 numbers flat bet with 1 unit each for the next 6 spins
7 numbers flat bet with 1 unit each for the next 5 spins
8 numbers flat bet with 1 unit each for the next 4 spins
9 numbers flat bet with 1 unit each for the next 4 spins
10 numbers flat bet with 1 unit each for the next 3 spins
11 numbers flat bet with 1 unit each for the next 3 spins

Eleven is the maximum numbers you could bet, this may happen if your first number was 17 for example, because 17 has 2 neighbours on the wheel's layout (like every number) and 8 neighbours regarding the table layout 13,14,15,16,18,19,20,21 (yes,the diagonals too).

What if someone is lucky enough and after adding all neighbours one by one continues to win?
In this case you start adding 1 unit each time you win to only 1 of your betting numbers, you should start from your first selection and continue with the same order.
The minimum total of betting numbers are 6, for example 34 has 2 neighbours at the wheel's layout (17,6) and 3 neighbours at the table (31,32,35)
All together with 34 are six numbers.

We have seen how it might proceed if you are lucky and win frequently, this is not so rare because sometimes the croupiers are hitting specific sectors/numbers frequently (more than their probability)
Personally I've reached two times the 6 numbers bet selection after not many trials, of course this is not always the case.
Let's see what happens when we eventually lose, when you lose during 2 numbers bet selection or more, then you just pick another number and start from scratch, which means flat bet 1 number for the next 36 spins.

There is only one exception, if you lose during 1 number bet selection, in that case you pick another number but this time the betting wouldn't be flat, each and every time our number fails to hit we would add some interest by adding 1 unit.
So the bet on our number would be like this:
1st spin 1 unit
2nd spin 2 units
3rd spin 3 units
4th spin 4 units and so on till your number eventually appears or till you have lost for 37 spins in a row.

If you lose for second consecutive time your 37 bet cycle,this means that somewhere ahead are 4 more winning cycles/rounds but you would only need 2 of them in order to overcome the 2 negative ones.
You change again the number and continue with the dynamic progression till your winning cycles/rounds equals your losing ones.

Let's calculate the losing and winning totals by betting with the dynamic progression instead of the flat bet.
Remember that the average expectation is to lose 13 times within 37 cycles/rounds, so the total loss will be:
[13 x (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37)]=>
13 x 703 = -9139 units

Now let's calculate the total amount of profit:

[(7 x 35) - 21] + [(8 x 35) - 28] + [(9 x 35) - 36] +[(10 x 35) - 45] + [(11 x 35) - 55] + [(12 x 35) - 66] + [(13 x 35) - 78] + [(14 x 35) - 91] + [(15 x 35) - 105] + [(16 x 35) - 120] + [(17 x 35) - 136] +[(18 x 35) - 153] + [(19 x 35) - 171] + [(20 x 35) - 190] + [(21 x 35) - 210] + [(22 x 35) - 231] + [(23 x 35) - 253] + [(24 x 35) - 276] + [(25 x 35) - 300] + [(26 x 35) - 325] + [(27 x 35) - 351] + [(28 x 35) - 378] + [(29 x 35) - 406] + [(30 x 35) - 435] =>

(245 - 21) + (280 - 28) + (315 - 36) + (350 - 45) + (385 - 55) + (420 - 66) + (455 - 78) + (490 - 91) + (525 - 105) + (560 - 120) + (595 - 136) + (630 - 153) + (665 - 171) + (700 - 190) + (735 - 210) + (770 - 231) + (805 - 253) + (840 - 276) + (875 - 300) + (910 - 325) + (945 - 351) + (980 - 378) + (1015 - 406) + (1050 - 435) =>

224 + 252 + 279 + 305 + 330 + 354 + 377 + 399 + 420 + 440 + 459 + 477 + 494 + 510 + 525 + 539 + 552 + 564 + 575 + 585 + 594 + 602 + 609 + 615 = +11080

We deduct 9139 from 11080 and we find 1941 net profit, therefore the negative expectation has been turned into positive!
But wait, there are more good news!

Since always the first losing cycle/round costs us 36 units instead of 703, that's why you don't need the exact probability to happen (2 winning cycles for every 1 losing cycle), you need approximately equal winning/losing cycles.
I say approximately because you never know in which spin your number is going to appear, thus the net gain differs.
Let me put it this way,if after 2 losing cycles you have 2 winning ones, then you are in profit and re-start from 1 number flat bet with 1 unit for 36 spins.

But after 3 or more consecutive losing cycles you would need the same amount of winning cycles plus 1 more.
Even in such situation, you can be in the positive with worst results than what probability theory supports.
For example in order to overcome 3 losing rounds, I just need 4 instead of 6 which probability dictates as average.
Of course,the results could also be better than average!:-)

Just remember that in the first phase, while we flat bet, we try to find the possibility to win by frequent repeaters.
We build slowly and safely a net gain and we don't limit the possibility for more profits (sky is the limit!) but we limit the loss.

During any stage of the 1st phase (flat bet) the maximum amount we risk is 36 units or less, if you don't lose during the 1st (1x36) or 2nd stage (2x18) of the first phase, then you would be in the positive.
The longer you keep on winning with the flat bets, the more would be the profit, it's all about what happens first.

Under any circumstance you will know what to do next, I'd recommend a total bankroll of 2000 minimum, 6000 average and 10000 units maximum in order to overcome occasional distribution anomalies.

Angelo A.

[Toby]

Blue Angel, a nice analisis

I saw some miss calculations

"This means that if someone was betting one number for 37 spins, 24 times he/she would have won and 13 times he/she would have lost."
The chance to hit 1 or more times in 35 spins(at least +1) is 61%(24hits is over 64%), the chance to hit 1or more  times in 36 spins is 62%, and 63% the chance to hit 1 or more in 37

"Those extremes are from my experience during gambling sessions and not from simulations."
Actual facts must be set with several thounsend of trials. Our bounderies 18-30 are short term.

Your ideas should be tested with large simulations. It would not be hard to determine how your system works.

[Blue_Angel]

Blue Angel, a nice analisis

I saw some miss calculations

"This means that if someone was betting one number for 37 spins, 24 times he/she would have won and 13 times he/she would have lost."
The chance to hit 1 or more times in 35 spins(at least +1) is 61%(24hits is over 64%), the chance to hit 1or more  times in 36 spins is 62%, and 63% the chance to hit 1 or more in 37

"Those extremes are from my experience during gambling sessions and not from simulations."
Actual facts must be set with several thounsend of trials. Our bounderies 18-30 are short term.

Your ideas should be tested with large simulations. It would not be hard to determine how your system works.

24 / 37 = 0.6486 x 100 = 64.86%

By the way is thousand, not ''thounsend''

So if you or any other wants to try it for thousands, or even millions of outcomes, be my guest.

[nottophammer]

Dear readers,

this is a thread which requires your full attention in order to fully comprehend the hidden opportunity.

Let's start with the basic about the Law Of the Third:

In every 37 spins cycle there are 24 numbers which have appeared once or more and 13 sleepers
(numbers which haven't appeared within the last 37 spins)
Those are average numbers and deviations exist, for example I've seen up to 30 different numbers to show up within the 37 last spins and the least were 18 different numbers.
Those extremes are from my experience during gambling sessions and not from simulations.
In order to find the average we should determine the extremes,or in other words the limits, in that case 18 and 30 are the limits.

18 19 20 21 22 23 24 26 26 27 28 29 30 start by discarding the outer numbers from both sides:

First discarding 18 and 30
Second discard 19 and 29
Third discard 20 and 28
Fourth discard 21 and 27
Fifth discard 22 and 26
Sixth discard 23 and 25

And we conclude to 24 which is the average total of the appeared numbers within 37 last spins.
Another way to calculate the average is to add all the totals and then divide the total sum with the total of the averages,for example:

18 different numbers within the last 37 spins happened 1 time
19 different numbers within the last 37 spins happened 2 times
20 different numbers within the last 37 spins happened 3 times
21 different numbers within the last 37 spins happened 4 times
22 different numbers within the last 37 spins happened 5 times
23 different numbers within the last 37 spins happened 6 times
24 different numbers within the last 37 spins happened 7 times
25 different numbers within the last 37 spins happened 6 times
26 different numbers within the last 37 spins happened 5 times
27 different numbers within the last 37 spins happened 4 times
28 different numbers within the last 37 spins happened 3 times
29 different numbers within the last 37 spins happened 2 times
30 different numbers within the last 37 spins happened 1 times

[(1 x 18)+(2 x 19)+(3 x 20)+(4 x 21)+(5 x 22)+(6 x 23)+(7 x 24)+(6 x 25)+(5 x 26)+(4 x 27)+(3 x 28)+(2 x 29)+(1 x 30) / 49] =>

[(18+38+60+84+110+138+168+150+130+108+84+58+30) / 49]=>

1176 / 49 = 24

This means that if someone was betting one number for 37 spins, 24 times he/she would have won and 13 times he/she would have lost.
To calculate the total sum of lost bets is easy:
13 times multiplied by 37 (1 x 37) equals minus 481 units.

In order to find the total amount of won bets, we should again calculate the averages as we did above,let's see:
We start from the middle numbers within a 37 spin cycle and add 2 more numbers from both sides till we have a total of 24.
18th and 19th spins
17th and 20th spins
16th and 21st spins
15th and 22nd spins
14th and 23rd spins
13th and 24th spins
12th and 25th spins
11th and 26th spins
10th and 27th spins
9th and 28th spins
8th and 29th spins
7th and 30th spins
So we have 24 wins from different spins, now let's calculate the total amount of net profit.

29+28+27+26+25+24+23+22+21+20+19+18+17+16+15+14+13+12+11+10+9+8+7+6 = 420

But since we would lose 481 - 420 = -61 loss, therefore the definition negative expectation.

Do we agree so far?
Is everything clear?

Let's see how we could turn the negative expectation into positive without changing a thing in terms of probability, in fact with exactly the same results which have lead us to the negative balance above.

The average expectation is to get twice the wins for every loss, with my method you only need approximately 1 positive cycle for every negative cycle, which means that the proportion of 2 to 1 changed to 1 to 1.

Let me explain how this is possible, the only thing which I'm not going to reveal you here is the criteria which I'm using to select the betting number. this is far as i have got,should i really read on as this part you are not going to show?will it end up like having a book with the important page missing
You could pick a random number, or your "lucky" number or anything else you like.

I start flat betting 1 number with 1 unit for 36 spins.
When the betting number appears, no matter in which spin, I re-bet the same number plus its neighbour regarding the wheel layout (right or left doesn't matter)

So now I'm flat betting 2 numbers with 1 unit each for the next 18 spins.
When one of the two betting numbers hits within eighteen spins, then I re-bet those two numbers and adding the other wheel neighbour, thus in total three numbers for the next 12 spins, always flat bet 1 unit each.

When one of the three betting numbers comes, I add one more neighbour,this time from the table layout.
So far we have 4 numbers to flat bet with 1 unit each for the next 9 spins.
Let's see if you keep on winning what happens:

5 numbers flat bet with 1 unit each for the next 7 spins
6 numbers flat bet with 1 unit each for the next 6 spins
7 numbers flat bet with 1 unit each for the next 5 spins
8 numbers flat bet with 1 unit each for the next 4 spins
9 numbers flat bet with 1 unit each for the next 4 spins
10 numbers flat bet with 1 unit each for the next 3 spins
11 numbers flat bet with 1 unit each for the next 3 spins

Eleven is the maximum numbers you could bet, this may happen if your first number was 17 for example, because 17 has 2 neighbours on the wheel's layout (like every number) and 8 neighbours regarding the table layout 13,14,15,16,18,19,20,21 (yes,the diagonals too).

What if someone is lucky enough and after adding all neighbours one by one continues to win?
In this case you start adding 1 unit each time you win to only 1 of your betting numbers, you should start from your first selection and continue with the same order.
The minimum total of betting numbers are 6, for example 34 has 2 neighbours at the wheel's layout (17,6) and 3 neighbours at the table (31,32,35)
All together with 34 are six numbers.

We have seen how it might proceed if you are lucky and win frequently, this is not so rare because sometimes the croupiers are hitting specific sectors/numbers frequently (more than their probability)
Personally I've reached two times the 6 numbers bet selection after not many trials, of course this is not always the case.
Let's see what happens when we eventually lose, when you lose during 2 numbers bet selection or more, then you just pick another number and start from scratch, which means flat bet 1 number for the next 36 spins.

There is only one exception, if you lose during 1 number bet selection, in that case you pick another number but this time the betting wouldn't be flat, each and every time our number fails to hit we would add some interest by adding 1 unit.
So the bet on our number would be like this:
1st spin 1 unit
2nd spin 2 units
3rd spin 3 units
4th spin 4 units and so on till your number eventually appears or till you have lost for 37 spins in a row.

If you lose for second consecutive time your 37 bet cycle,this means that somewhere ahead are 4 more winning cycles/rounds but you would only need 2 of them in order to overcome the 2 negative ones.
You change again the number and continue with the dynamic progression till your winning cycles/rounds equals your losing ones.

Let's calculate the losing and winning totals by betting with the dynamic progression instead of the flat bet.
Remember that the average expectation is to lose 13 times within 37 cycles/rounds, so the total loss will be:
[13 x (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37)]=>
13 x 703 = -9139 units

Now let's calculate the total amount of profit:

[(7 x 35) - 21] + [(8 x 35) - 28] + [(9 x 35) - 36] +[(10 x 35) - 45] + [(11 x 35) - 55] + [(12 x 35) - 66] + [(13 x 35) - 78] + [(14 x 35) - 91] + [(15 x 35) - 105] + [(16 x 35) - 120] + [(17 x 35) - 136] +[(18 x 35) - 153] + [(19 x 35) - 171] + [(20 x 35) - 190] + [(21 x 35) - 210] + [(22 x 35) - 231] + [(23 x 35) - 253] + [(24 x 35) - 276] + [(25 x 35) - 300] + [(26 x 35) - 325] + [(27 x 35) - 351] + [(28 x 35) - 378] + [(29 x 35) - 406] + [(30 x 35) - 435] =>

(245 - 21) + (280 - 28) + (315 - 36) + (350 - 45) + (385 - 55) + (420 - 66) + (455 - 78) + (490 - 91) + (525 - 105) + (560 - 120) + (595 - 136) + (630 - 153) + (665 - 171) + (700 - 190) + (735 - 210) + (770 - 231) + (805 - 253) + (840 - 276) + (875 - 300) + (910 - 325) + (945 - 351) + (980 - 378) + (1015 - 406) + (1050 - 435) =>

224 + 252 + 279 + 305 + 330 + 354 + 377 + 399 + 420 + 440 + 459 + 477 + 494 + 510 + 525 + 539 + 552 + 564 + 575 + 585 + 594 + 602 + 609 + 615 = +11080

We deduct 9139 from 11080 and we find 1941 net profit, therefore the negative expectation has been turned into positive!
But wait, there are more good news!

Since always the first losing cycle/round costs us 36 units instead of 703, that's why you don't need the exact probability to happen (2 winning cycles for every 1 losing cycle), you need approximately equal winning/losing cycles.
I say approximately because you never know in which spin your number is going to appear, thus the net gain differs.
Let me put it this way,if after 2 losing cycles you have 2 winning ones, then you are in profit and re-start from 1 number flat bet with 1 unit for 36 spins.

But after 3 or more consecutive losing cycles you would need the same amount of winning cycles plus 1 more.
Even in such situation, you can be in the positive with worst results than what probability theory supports.
For example in order to overcome 3 losing rounds, I just need 4 instead of 6 which probability dictates as average.
Of course,the results could also be better than average!:-)

Just remember that in the first phase, while we flat bet, we try to find the possibility to win by frequent repeaters.
We build slowly and safely a net gain and we don't limit the possibility for more profits (sky is the limit!) but we limit the loss.

During any stage of the 1st phase (flat bet) the maximum amount we risk is 36 units or less, if you don't lose during the 1st (1x36) or 2nd stage (2x18) of the first phase, then you would be in the positive.
The longer you keep on winning with the flat bets, the more would be the profit, it's all about what happens first.

Under any circumstance you will know what to do next, I'd recommend a total bankroll of 2000 minimum, 6000 average and 10000 units maximum in order to overcome occasional distribution anomalies.

Angelo A.

[Blue_Angel]

At least you are one of the smart folks because you realise the importance of selection, choosing right from wrong makes the whole difference between winners and losers, if every selection was the same then what's the point to speak about prediction accuracy??

However, nobody would ever reveal you the whole solution, but just parts and for the savvy would be sufficient in order to find their way through the rest of the steps.

Here is the "raw" version, but I've fine tuned it by applying a tweak and the selection criteria for picking numbers with better than average probability.
Just a sidenote, the selection method for picking numbers alone could win with flat bets in long term.
I'm willing to prove you this kind of accuracy but without revealing the steps of the method.

[dimsun]

However, nobody would ever reveal you the whole solution, but just parts and for the savvy would be sufficient in order to find their way through the rest of the steps.

Not reveal whole solution, never proved wrong if fail. 

[Blue_Angel]

By the way, I was reading the topic "DNA of roulette" and I've to admit that I failed to see what betting 1 dozen and 1 column has to do with "law of the thirds", disappointment in my opinion.

The so called "hot" and "cold" numbers never belong to only 1 sector of the wheel's or table's layout, they belong to more than 1 groups/sectors, this is a very common fallacy, to consider numbers in groups just because of their proximity regarding those 2 layouts.
These 2 layouts (wheel's & table's) are only 2 out of many other possible layouts, just imagine how differently you could consider outcomes when you release your perception out of the mainstream frames!

[Blue_Angel]

Not reveal whole solution, never proved wrong if fail. 

Regardless of what one might think, I don't have to prove anything except to myself.

[denzie]

Is this as good as your other holy grail a while back ?

[Blue_Angel]

Is this as good as your other holy grail a while back ?

much better than your parlay on 2 dozens/columns

[denzie]

much better than your parlay on 2 dozens/columns

I parlay ec's but who cares right. (And I hate columns ).

Anyway carry on 

[denzie]

Not reveal whole solution, never proved wrong if fail. 

Yup

[falkor2k15]

There is only one exception, if you lose during 1 number bet selection, in that case you pick another number but this time the betting wouldn't be flat, each and every time our number fails to hit we would add some interest by adding 1 unit.
So the bet on our number would be like this:
1st spin 1 unit
2nd spin 2 units
3rd spin 3 units
4th spin 4 units and so on till your number eventually appears or till you have lost for 37 spins in a row.

If you lose for second consecutive time your 37 bet cycle,this means that somewhere ahead are 4 more winning cycles/rounds but you would only need 2 of them in order to overcome the 2 negative ones.
You change again the number and continue with the dynamic progression till your winning cycles/rounds equals your losing ones.

Let's calculate the losing and winning totals by betting with the dynamic progression instead of the flat bet.
Remember that the average expectation is to lose 13 times within 37 cycles/rounds, so the total loss will be:
[13 x (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37)]=>
13 x 703 = -9139 units

You need more than 10,000 to play this!? 

[Blue_Angel]

You need more than 10,000 to play this!? 

Assuming you just randomly picking numbers and just change number after a losing cycle, you'll need roughly 3000 units.

Without my extra tweak and selection criteria, you should expect more intensive fluctuations,  such as 4 losing cycle in a row.
I'd not recommend to anyone trying this method with less than 3000 units, this is the minimum bankroll requirement.

Yes it is very bankroll demanding, yes it is like putting too many eggs inside one basket!
That's why is not for everyone, also you need clear mind, the balls of a bull and nerves of steel to apply it properly!
Of course the potential profit compensates for everything else.

[Blue_Angel]

I parlay ec's but who cares right. (And I hate columns ).

Anyway carry on 

Just a friendly advice, EC streaks are choppier than 2 dozens streaks.

So if you are looking for parlaying, you are looking in the wrong place.