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Hidden opportunities within abbr. wheels

Topic closed. 55 replies. Last post 12 years ago by lottoarchitect.

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Greece
Member #2815
November 18, 2003
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Posted: December 5, 2004, 11:58 am - IP Logged

Well, here is an important matter regarding wheels that never discussed before. I've come up with this observation after I constructed a wheel optimiser to generate wheels and I faced an unexpected behaviour. I'll post here some results, so here we go.



Everyone knows what is an abbreviated wheel (either close or open cover). If we want a guarantee e.g. 2if3 in 10 balls (for a hypothetical jackpot 3 balls game), then we pick a wheel that offers 100% 2if3 with the desired number of balls to wheel. The above example needs 8 tickets for that.

The BIG question is: what if we don't match the condition 3 correct out of 10; what if we we match only 2? Still the wheel offers chances to win in 2if2, althought most of the time they are reduced. Here is a comparison example of two similar wheels (type & size) that reflect a very important aspect of wheels, most people never discussed about.



1st wheel C(3,10,2,3)=8

1 - 1 5 9

2 - 1 6 7

3 - 2 3 8

4 - 2 4 8

5 - 2 8 10

6 - 3 4 10

7 - 5 6 7

8 - 6 7 9



2nd wheel C(3,10,2,3)=8

1 - 1 2 10

2 - 1 3 8

3 - 1 4 6

4 - 2 3 4

5 - 2 6 8

6 - 3 6 10

7 - 4 8 10

8 - 5 7 9



(BobP, this is a split wheel and a better one strangely!)



Now, we get the coverage information of the above wheels:



1st wheel coverage rates



T if M    Tested Covered        % Not Covered     %

----------------------------------------------------

2 If 2 :      45      20 44.44444      25 55.55556

2 If 3 :     120     120 100.00000       0   0.00000

2 If 4 :     210     210 100.00000       0   0.00000

2 If 5 :     252     252 100.00000       0   0.00000

2 If 6 :     210     210 100.00000       0   0.00000

2 If 7 :     120     120 100.00000       0   0.00000

3 If 4 :     210      50 23.80952     160 76.19048

3 If 5 :     252     132 52.38095     120 47.61905

3 If 6 :     210     174 82.85714      36 17.14286

3 If 7 :     120     120 100.00000       0   0.00000



2nd wheel coverage rates



T if M    Tested Covered        % Not Covered     %

----------------------------------------------------

2 If 2 :      45      24 53.33333      21 46.66667

2 If 3 :     120     120 100.00000       0   0.00000

2 If 4 :     210     210 100.00000       0   0.00000

2 If 5 :     252     252 100.00000       0   0.00000

2 If 6 :     210     210 100.00000       0   0.00000

2 If 7 :     120     120 100.00000       0   0.00000

3 If 4 :     210      56 26.66667     154 73.33333

3 If 5 :     252     147 58.33333     105 41.66667

3 If 6 :     210     189 90.00000      21 10.00000

3 If 7 :     120     120 100.00000       0   0.00000



Now, as you can see, the second wheel offers better chances to win at 2if2. The second wheel is an optimized one. So, even if both wheels offer 100% chance to win in 2if3, however the 2nd one offers around 9% better chances to win in 2if2, in case we do not hit 3 numbers correct but only 2 out of 10. Similar is the observation for the other categories too. Obviously, the 2nd wheel is much better than the 1st one. As you understand, the optimization refers to the need of improving all other categories (hidden opportunities) besides the initial category of the wheel (or even the initial category in case of open-cover wheels). The reason is simple enough: we want to win and because we cannot guarantee that our selection of numbers is always correct (e.g. hit 3 out of 10), we want to have better chances if other cases occur too. What do you think of this matter?

If you have something to do, at least do it well...

    BobP's avatar - bobp avatar.png
    Dump Water Florida
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    Posted: December 5, 2004, 1:03 pm - IP Logged

    Both wheels are split in the sense every digit doesn't come in contact with every other digit as each appears a maximum of three times and can come in contact with a maximum of six other digits.  I do take your meaning in the second as 579 is exclusive.

    I'm not surprised the split wheel has better lower tier coverage, they usually do which is why they lure people into playing them, but the jackpot then becomes more elusive.  Too bad there is no 2if3 prize in Pick-3 we'd win two draws out of three or make a 2if2 wheel and win almost every time.

    I agree the better the coverage of everything the better, provided we don't pay a price to get it.  A better made wheel is just that. 

    BobP
     

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      Greece
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      Posted: December 5, 2004, 1:19 pm - IP Logged

      The important thing I illustrate above is that we have two wheels with the exact same attributes (guarantee the same prize and both need the same amount of tickets). We can see that they have different properties in terms of coverage on other prize divisions.

      Sadly, no jackpot 3 games exist but similar is the observation I have on many jackpot 6 game wheels, where more lower divisions occur. As for the jackpot, we already know it depends directly on the total tickets played and not at the properties of each wheel.

      If you have something to do, at least do it well...

        RJOh's avatar - chipmunk
        mid-Ohio
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        Posted: December 5, 2004, 2:16 pm - IP Logged

        Quote: Originally posted by apagogeas on December 5, 2004




        Everyone knows what is an abbreviated wheel (either close or open cover). If we want a guarantee e.g. 2if3 in 10 balls (for a hypothetical jackpot 3 balls game), then we pick a wheel that offers 100% 2if3 with the desired number of balls to wheel. The above example needs 8 tickets for that.




        If a 3 of 10 lottery existed, the odds of winning would be:
        Match   odds
        3/3        1:120
        2/3        1:6
        1/3        1:2
        overall    1:1.4

        How would it ever generate any profit and a jackpot worth winning? And what would be the point of playing 8 lines to match 2of3?

        RJOh

         * you don't need to buy more tickets, just buy a winning ticket * 
           
                     Evil Looking       

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          Greece
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          Posted: December 5, 2004, 2:40 pm - IP Logged

          RJOh,

          First of all, we do not use an abbreviated wheel to hit the jackpot. Chances to hit the jackpot are directly related to the total tickets played as any other method of generating tickets to play.

          Second, abbreviated wheels are good if we want to ensure a guarantee for a lesser prize category with the minimal possible cost. This is impossible with quick picks or if you randomly arrange your numbers. You'll end up with many more tickets to ensure that guarantee.

          As for the Pick 3 example, I just illustrated what's going on with several wheels' properties unseen by many. This optimization can be applied to any wheel (pick 6/7 etc). Improvements can be found anywhere as I have seen so far. Also, if we use this 2if3 wheel for pick 5 lotteries (using 2 banker numbers->12 numbers to play in total), we still have improved chances to hit with only 8 tickets. So this wheel is not that useless as you may think. The point is that we have increased chances to win if we use an optimized wheel, even if we do not match 3 out of 10 but only 2 out of 10. This is the purpose of the thread I posted here.

          If you have something to do, at least do it well...

            RJOh's avatar - chipmunk
            mid-Ohio
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            Posted: December 5, 2004, 4:09 pm - IP Logged

            As I noted in my post, the odds of matching 2of3 in a 3/10 game would be 1:6, so why would playing 8 lines be a good strategy?  Maybe the optimization of this strategy might be clearer if you explained it using a 6/49 game where the odds of matching 2of6 is 1:8.

            RJOh

             * you don't need to buy more tickets, just buy a winning ticket * 
               
                         Evil Looking       

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              Greece
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              Posted: December 5, 2004, 4:14 pm - IP Logged

              I'm not talking about strategies here RJOh.

              I'm talking about improving wheels. This is totally different that what you are saying. Don't confuse the optimization of wheels with strategies.

              A strategy is to use a wheel to play. However, it is preferable to use an optimized wheel (if you choose to use wheels) rather than an arbitrary similar wheel with the same attributes, as I try to illustrate in this thread. I just presented an example of a pick 3 optimized wheel here, that's all. People who use wheels can understand the importance of this matter and I ask their opinion to this.

              If you have something to do, at least do it well...

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                Posted: December 5, 2004, 5:41 pm - IP Logged

                In Maryland we can play only two numbers but they have to be either front pair or back pair.  For $.50 I believe it's $25.00 and $1.00 bet is $50.00.  Oh the pairs have to be in picked order no boxes.  Just a thought while you play with the wheels.

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                  Greece
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                  Posted: December 5, 2004, 5:46 pm - IP Logged

                  A misunderstanding Maryland. The above Pick 3 wheel is for jackpot games, no cash 3 etc. We are not talking for boxed neither straight wheels here. These are not abbreviated wheels and the above analysis does not apply to that sort of wheels.

                  If you have something to do, at least do it well...

                    Avatar
                    Melbourne
                    Australia
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                    Posted: December 5, 2004, 7:01 pm - IP Logged

                    Well, it seems that if we optimise our wheels then we get an improved second chance if our predictions are a bit off. I see from the Lotto Architect website that you intend to implement this tool into the program at some future stage. Is this the only way this tool will be made available? Whatever, bring it on I say. Looks like this is good for LA users.



                    And whilst not a mathematician, I note also at the website you refer to world records being equalled if not beaten with this tool. Have world records in the past only been recognised because of minimum lines in the wheel? For if this is the case and this tool can now offer both minumum lines plus an improved second chance then I have to say this is pretty revolutionary stuff.



                    regards



                    relowe

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                      Greece
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                      Posted: December 5, 2004, 7:55 pm - IP Logged

                      Quite correct relowe, this is the main idea of an optimized wheel, assuming we cannot reduce the required tickets even further or improve the coverage of the main category in case of an open-cover wheel.

                      As for the equality to world records, it has managed to match all world record wheels I've tested for up to 40-60 tickets and also improved several of them (these cannot be reduced further). I haven't tested it on big wheels as I have to improve the speed even more for that but I can say it has good chances to break a world record if applied to large ticket wheels, as those have still room for improvement. And yes, world records are based only to total tickets included in the wheel and not their 'hidden properties' like the one I mentioned in this thread.

                      If you have something to do, at least do it well...

                        BobP's avatar - bobp avatar.png
                        Dump Water Florida
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                        Posted: December 6, 2004, 12:55 am - IP Logged

                        I don't know about you, but every ticket I buy is in the hope of hitting a jackpot no matter what the wheel guarantee happens to be.

                        The strategy is the hope the wheel will play above it's guarantee.  To have three or four numbers guaranteed on a ticket when conditions are met and then to have two or three numbers more correct on that same line.

                        I win lot's of 3# prizes, why can't I ever seem to get two of them on the same line?

                        Anyway, if you examine the hand made "perfect" wheels lottery experts made in the old days there is a reason they can often be further optimized.

                        What you find is a pattern like this inside a (18,6,4,6) or (24,6,4,6)
                        1-2-3-4-5-6
                        1-2-3-6-7-8
                        1-2-3-7-8-9
                        1-2-3-10-11-12
                        1-2-3-13-14-15
                        1-2-3-16-17-18
                        4-5-6-7-8-9
                        4-5-6-10-11-12
                        4-5-6-13-14-15
                        4-5-6-16-17-18
                        7-8-9-10-11-12
                        7-8-9-13-14-15
                        7-8-9-16-17-18
                        10-11-12-13-14-15
                        10-11-12-16-17-18
                        13-14-15-16-17-18

                        What you have is a 3+3=JP group wheel inside the 4if6 wheel where you still win the 4if6 prize, but have a jackpot shot by hand selecting the positions of your numbers as groups you hope will hit together in groups of three, two groups correct = Jackpot.  One group and 2 in another = 5# prize, even one groups correct = multiple 4# prizes when all six winning numbers fall among the numbers wheeled.

                        There is more then just optimization to make the best wheel for lotto.

                        Here is an example 48 lines to guarantee 4if6in18 numbers which usually can be done in 42, but this wheel contains a 3+3 jackpot group wheel and 100% 3if3 coverage in case you don't get all six winning numbers.  Wheels usually cover one or the other guarantee, not both for your money.  Would cost over a hundred lines to get all three wheels independently.  Please consider this wheel copyrighted. 

                         1  2  3  4  5  6
                         1  2  3  7  8  9
                         1  2  3 10 11 12
                         1  2  3 13 14 15
                         1  2  3 16 17 18
                         1  4  7 12 14 17
                         1  4  8 11 15 16
                         1  4  9 10 13 18
                         1  5  7 12 15 18
                         1  5  8 11 13 17
                         1  5  9 10 14 16
                         1  6  7 12 13 16
                         1  6  8 11 14 18
                         1  6  9 10 15 17
                         2  4  7 11 13 18
                         2  4  8 10 14 17
                         2  4  9 12 15 16
                         2  5  7 11 14 16
                         2  5  8 10 15 18
                         2  5  9 12 13 17
                         2  6  7 11 15 17
                         2  6  8 10 13 16
                         2  6  9 12 14 18
                         3  4  7 10 15 16
                         3  4  8 12 13 18
                         3  4  9 11 14 17
                         3  5  7 10 13 17
                         3  5  8 12 14 16
                         3  5  9 11 15 18
                         3  6  7 10 14 18
                         3  6  8 12 15 17
                         3  6  9 11 13 16
                         4  5  6  7  8  9
                         4  5  6 10 11 12
                         4  5  6 13 14 15
                         4  5  6 16 17 18
                         7  8  9 10 11 12
                         7  8  9 13 14 15
                         7  8  9 16 17 18
                        10 11 12 13 14 15
                        10 11 12 16 17 18
                        13 14 15 16 17 18
                         1  2  7  8 10 12
                         1  3  7  9 11 12
                         2  3  8  9 10 11
                         4  5 13 14 16 18
                         4  6 13 15 17 18
                         5  6 14 15 16 17
                        end 48 lines 100% 4if6, 3+3, 3if3,  coverage in 18 numbers. BobP

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                          Greece
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                          Posted: December 6, 2004, 7:24 am - IP Logged

                          BobP,

                          this C(6,18,4,6)=48 is a nice wheel for the purpose it is designed for. However it is unfair to compare it with the WR or an optimized one which has only 42 lines.

                          As for the jackpot, everyone wants to hit the jackpot and every wheel has a chance on that, no matter how it is constructed or its guarantee. Even in your 48 ticket wheel, the 3+3 matter for hitting the jackpot is not that important. These are the lines of your wheel that you can setup in the 3+3 manner to hit the jackpot (groups 1-3,4-6,7-9,10-12,13-5,16-18),

                          1 2 3 4 5 6

                          1 2 3 7 8 9

                          1 2 3 10 11 12

                          1 2 3 13 14 15

                          1 2 3 16 17 18

                          4 5 6 7 8 9

                          4 5 6 10 11 12

                          4 5 6 13 14 15

                          4 5 6 16 17 18

                          7 8 9 10 11 12

                          7 8 9 13 14 15

                          7 8 9 16 17 18

                          10 11 12 13 14 15

                          10 11 12 16 17 18

                          13 14 15 16 17 18

                          15 lines in total. You lose control of the remaining 33 lines. Now, if you are lucky and have all six numbers in any two of these groups, then you have the jackpot.



                          The C(6,42,4,6)=42 wheel is

                          1 2 3 4 5 6

                          1 2 3 7 8 9

                          1 2 3 10 11 12

                          1 2 3 13 14 15

                          1 2 3 16 17 18

                          1 4 7 10 13 16

                          1 4 8 11 14 17

                          1 4 9 12 15 18

                          1 5 7 11 14 18

                          1 5 8 12 15 16

                          1 5 9 10 13 17

                          1 6 7 12 15 17

                          1 6 8 10 13 18

                          1 6 9 11 14 16

                          2 4 7 12 13 18

                          2 4 8 10 14 16

                          2 4 9 11 15 17

                          2 5 7 10 14 17

                          2 5 8 11 15 18

                          2 5 9 12 13 16

                          2 6 7 11 15 16

                          2 6 8 12 13 17

                          2 6 9 10 14 18

                          3 4 7 11 13 17

                          3 4 8 12 14 18

                          3 4 9 10 15 16

                          3 5 7 12 14 16

                          3 5 8 10 15 17

                          3 5 9 11 13 18

                          3 6 7 10 15 18

                          3 6 8 11 13 16

                          3 6 9 12 14 17

                          4 5 6 7 8 9

                          4 5 6 10 11 12

                          4 5 6 13 14 15

                          4 5 6 16 17 18

                          7 8 9 10 11 12

                          7 8 9 13 14 15

                          7 8 9 16 17 18

                          10 11 12 13 14 15

                          10 11 12 16 17 18

                          13 14 15 16 17 18



                          Same as above, the 3+3 setup can work on the following lines.

                          1 2 3 4 5 6

                          1 2 3 7 8 9

                          1 2 3 10 11 12

                          1 2 3 13 14 15

                          1 2 3 16 17 18

                          4 5 6 7 8 9

                          4 5 6 10 11 12

                          4 5 6 13 14 15

                          4 5 6 16 17 18

                          7 8 9 10 11 12

                          7 8 9 13 14 15

                          7 8 9 16 17 18

                          10 11 12 13 14 15

                          10 11 12 16 17 18

                          13 14 15 16 17 18

                          We have 15 lines to do the same 3+3 trick here. We lose control of the 27 remaining lines only.

                          Still, nobody says that we can have all our winning numbers in any two groups. This simply means, even the ticket 2 6 9 10 14 18 has equal chance to be the jackpot ticket. The only draw back of the WR wheel above is that is does offer only 93.28% chance for 3if3 (I haven't tried to optimise it yet).

                          In terms of the jackpot, your wheel has 48/C(49,6)=0.000343% whilst the WR wheel has 42/C(49,6)=0.0003% chances to hit.

                          In case all 6 numbers are among the 18, then

                          WR=0.2262%

                          48 ticket wheel=0.2585%

                          No big difference whichever wheel we use, don't you think?



                          In either case, the jackpot is not something that should bother you. If it comes, you are lucky but it is not the crucial part of choosing an abbreviated wheel to play. This is what I'm saying all the time. Your wheel simply offers a better 'phenomenal' way to hit the jackpot IN CASE all 6 winning numbers fall within 2 groups only, as it does the WR wheel too.

                          Your increase of cost is 48/42=1.1428 or we have a free WR wheel play for every 7 plays of your 48 ticket wheel (8 plays for WR wheel at the cost of 7).



                          If 3 numbers match among 18, then WR record wheel offers

                               6         5         4         3           %       Total       Acc %   

                          --------------------------------------------------------------------------------------

                               -         -         -         -          54     6.61765     6.61765

                               -         -         -         1         729    89.33824    95.95588

                               -         -         -         3          27     3.30882    99.26471

                               -         -         -         5           6     0.73529   100.00000



                          48 ticket wheel offers

                               6         5         4         3           %       Total       Acc %   

                          --------------------------------------------------------------------------------------

                               -         -         -         1         720    88.23529    88.23529

                               -         -         -         2          72     8.82353    97.05882

                               -         -         -         3          12     1.47059    98.52941

                               -         -         -         5          12     1.47059   100.00000



                          Besides the 100% 3if3 guarantee the 48 ticket wheel offers, as you can see, the 6.62% loss of the WR wheel is easily covered by the free play we have. There is a slight increase of odds for more 3 hits but this is due to the increase of tickets and also it does a bit better in case of 4 or 5 out of 18 match as expected.

                          I'm not trying to lower the quality of this 48 ticket wheel for sure. I just cannot see a good reason to go for the 48 ticket wheel when the 3if3 wins cannot cover the costs on both wheels, the 3+3 trick is controlled totally by luck (and can be done on both wheels) and the jackpot chances are almost the same on both wheels.

                          If you have something to do, at least do it well...

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                            Greece
                            Member #2815
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                            Posted: December 6, 2004, 7:58 am - IP Logged

                            And finally, there is another optimization I haven't talked about yet. It refers to optimizing multiple hits with the increase of coverage. There, we put our numbers in order of preference and if they hit then we have maximum outcome. No need for groups, 3+3 tactics etc. Of course, the wheel retains its initial guarantee, no matter which numbers are finally drawn. It is just a better way to have the player control his luck by selecting the numbers he feels have more chances to come among the required wheel's numbers. Jackpot chances are still equal for all wheels of the same size (optimization cannot improve jackpot chances).

                            If you have something to do, at least do it well...

                              BobP's avatar - bobp avatar.png
                              Dump Water Florida
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                              Posted: December 7, 2004, 2:37 am - IP Logged

                              You might be missing the point of the thing. 

                              You open Lot book to 3if3of6in18 and it takes 48 lines with no 4if6 guarantee. 

                              You check 4if6in18 and the 42 lines offer no 3if3 guarantee. 

                              Neither offers a clear jackpot shot for those inclined, they'd have to map the wheel to do it.  I've combined three wheels needing over one hundred lines into one 48 line wheel at no additional cost to the 3if3 wheel user and only six more lines for the 4if6 wheeler to hedge his bet as 5 winning numbers don't guarantee 4if6 but the 3if3 portion does guarantee a bigger pile of 3# winning tickets. 

                              In part the question is whether you'd like to take a hand in predicting the outcome with straight jackpot shots backed up by the same guarantee you'd have anyway, or just trust to luck.  The wheel works either way.

                              BobP