Wheel Analysis and Discussion

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draughtsman
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Wheel Analysis and Discussion

Post by draughtsman » Sun Jan 27, 2008 7:40 am

TRANSFERED FROM OLD FORUM

I have this question in my head for a long time but I'd like to share this with you to comment if you want. I'll concentrate on an example and I have picked the C(13,5,3,4)=16 wheel to base my analysis.

The C(13,5,3,4)=16 is an L=1 wheel which means we are guaranteed to win at least 1 #3-hit if we have at least 4 correct numbers selected. So far so good. What if we want to still pick only 13 numbers and let's say for simplicity we want to play double the lines (i.e. 32 tickets)? Definitely what we look for is an overlapping version of a wheel, where more hits will be produced. The real question I want to discuss about is what is the best way to increase the lines to play.

Solution 1: play the same tickets of the wheel twice in our example. Hmmm... that's a direct approach and does provide a wheel with at least L=2. Actually the C(13,5,3,4,L=2)=26 (I think) but that's another story.
So what we get by doing so?

Coverage information - 2xC(13,5,3,4)=16 ->32 tickets


CODE

Coverage information
------ ---------- ---------- ---------- ---------- ----------
P M Tested Covered Covered% Uncovered Uncovered%
------ ---------- ---------- ---------- ---------- ----------
2 2 78 78 100.00000% 0 0.00000%
2 3 286 286 100.00000% 0 0.00000%
2 4 715 715 100.00000% 0 0.00000%
2 5 1287 1287 100.00000% 0 0.00000%
2 6 1716 1716 100.00000% 0 0.00000%
2 7 1716 1716 100.00000% 0 0.00000%
3 3 286 154 53.84615% 132 46.15385%
3 4 715 715 100.00000% 0 0.00000%
3 5 1287 1287 100.00000% 0 0.00000%
3 6 1716 1716 100.00000% 0 0.00000%
3 7 1716 1716 100.00000% 0 0.00000%
4 4 715 84 11.74825% 631 88.25175%
4 5 1287 636 49.41725% 651 50.58275%
4 6 1716 1567 91.31702% 149 8.68298%
4 7 1716 1716 100.00000% 0 0.00000%
5 5 1287 17 1.32090% 1270 98.67910%
5 6 1716 135 7.86713% 1581 92.13287%
5 7 1716 457 26.63170% 1259 73.36830%



and an analysis of 4-hit category


CODE

Details information on 4 hits
------------------------------------------------------------------------
5 4 3 2 1 0 Total %
------------------------------------------------------------------------
0 3 2 14 9 4 1 0.14%
0 2 0-5 6-17 8-21 0-7 74 10.35%
0 1 0-6 6-16 9-21 0-7 9 1.26%
0 0 9 2-10 7-21 0-6 6 0.84%
0 0 8 4-9 10-18 2-5 5 0.70%
0 0 7 4-12 8-16 1-6 23 3.22%
0 0 6 3-13 7-23 0-7 82 11.47%
0 0 5 6-16 5-21 0-7 58 8.11%
0 0 4 7-17 5-20 0-7 238 33.29%
0 0 3 10-20 3-19 0-7 44 6.15%
0 0 2 13-21 3-17 0-7 164 22.94%
0 0 1 14-21 3-17 0-8 11 1.54%
0h 715 631 0 0 0 0 715
0% 100.00% 88.25% 0.00% 0.00% 0.00% 0.00%



Ok, not bad for sure. Now let's try to make a wheel from scratch that aims to produce a 32 ticket version of C(13,5,3,4,L>=2)=32

Solution 2: build an explicit C(13,5,3,4,L>=2)=32.

Coverage information


CODE

Coverage information
------ ---------- ---------- ---------- ---------- ----------
P M Tested Covered Covered% Uncovered Uncovered%
------ ---------- ---------- ---------- ---------- ----------
2 2 78 78 100.00000% 0 0.00000%
2 3 286 286 100.00000% 0 0.00000%
2 4 715 715 100.00000% 0 0.00000%
2 5 1287 1287 100.00000% 0 0.00000%
2 6 1716 1716 100.00000% 0 0.00000%
2 7 1716 1716 100.00000% 0 0.00000%
3 3 286 222 77.62238% 64 22.37762%
3 4 715 715 100.00000% 0 0.00000%
3 5 1287 1287 100.00000% 0 0.00000%
3 6 1716 1716 100.00000% 0 0.00000%
3 7 1716 1716 100.00000% 0 0.00000%
4 4 715 156 21.81818% 559 78.18182%
4 5 1287 937 72.80497% 350 27.19503%
4 6 1716 1700 99.06760% 16 0.93240%
4 7 1716 1716 100.00000% 0 0.00000%
5 5 1287 32 2.48640% 1255 97.51360%
5 6 1716 252 14.68531% 1464 85.31469%
5 7 1716 782 45.57110% 934 54.42890%



and an analysis of 4-hit category


CODE

Details information on 4 hits
------------------------------------------------------------------------
5 4 3 2 1 0 Total %
------------------------------------------------------------------------
0 2 0-2 13-16 10-15 1-4 4 0.56%
0 1 1-6 5-17 7-22 0-5 152 21.26%
0 0 7 5-11 9-19 1-5 8 1.12%
0 0 6 7-14 6-17 1-6 49 6.85%
0 0 5 7-15 8-19 1-6 106 14.83%
0 0 4 8-17 6-17 0-6 158 22.10%
0 0 3 10-19 6-17 1-6 174 24.34%
0 0 2 13-18 7-15 1-6 64 8.95%
0h 715 559 0 0 0 0 715
0% 100.00% 78.18% 0.00% 0.00% 0.00% 0.00%



Obviously both solutions are acceptable. Both are also L>=2 wheels. What makes a big difference however, is the coverage and hits achieved. On solution 1 wheel, we have only 53.85% coverage for the 3if3 category. Solution 2 offers 77.62% coverage. A good improvement at the same tickets. Similarly for the 4if4 category, solution 1 offers 11.75% cover and solution 2 offers 21.82%.
Also, when comparing the 4-hits tables of both wheels, we see that solution 1 has 88.25% NOT to hit a #4 success (rows 0h & 0%). Solution 2 reduces this to 78.18% NOT to hit a #4 success. So again, for the same amount of tickets, we have increased our chances to make a #4-hit by around 10%. Definitely not bad.

However, solution 2 lacks top #4 hits as the next comparison illustrates, but on the other hand gives more #4 hits in overall. We almost double our chances using solution 2 to make a #4-hit but they are not so glamorous as in solution 1.


CODE

------------------------------------------------------------------------
5 4 3 2 1 0 Total %
------------------------------------------------------------------------
Sol1> 0 3 2 14 9 4 1 0.14%
0 2 0-5 6-17 8-21 0-7 74 10.35%
0 1 0-6 6-16 9-21 0-7 9 1.26%
Sol2> 0 2 0-2 13-16 10-15 1-4 4 0.56%
0 1 1-6 5-17 7-22 0-5 152 21.26%



On the other end, at the lower hits we can see that solution 2 has reduced the chances for bad hits.


CODE

------------------------------------------------------------------------
5 4 3 2 1 0 Total %
------------------------------------------------------------------------
Sol1> 0 0 4 7-17 5-20 0-7 238 33.29%
0 0 3 10-20 3-19 0-7 44 6.15%
0 0 2 13-21 3-17 0-7 164 22.94%
0 0 1 14-21 3-17 0-8 11 1.54%
Sol2> 0 0 4 8-17 6-17 0-6 158 22.10%
0 0 3 10-19 6-17 1-6 174 24.34%
0 0 2 13-18 7-15 1-6 64 8.95%



What we can see above is that solution 1 generally will produce 1-4 #3-hits around 62.38% of the time whilst solution 2 will do that around 55.39% of the time. So we can say solution 2 reduces "bad" results by around 62.38-55.39=7%. This of course means more hits for solution 2 in better categories. So the question is what is better to do; produce bigger wheels from smaller ones such as solution 1 or make an exclusive build such as solution 2 for the targeting tickets size which seems to lack top hits but also tends to reduce "bad" hits?

What do you think?

lottoarchitect

I have noticed an error in the solution 1 #4-hits table (it should show 2 #3-hits minimum) but either way the conclusions are the same.

Hi LottoArchitect

Just a quick question before responding in more detail.

In your forum under Wheel Definitions you have a category Called "Other Wheels" in which you state, in part, the following:-

"...... Such wheels can be based on e.g. triple hits (organize numbers in groups) ....."

Is this part of the above, or can it be included in the above or, is it a separate issue all together and, therefore, "mutually exclusive?"

The reason behind my question is the need to remove the "very unlikely" combinations (e.g. 1-2-3-4-5 in a Pick 5 game) that wheels so often generate and substitute "more likely combinations.

I will be out of town for few days so will respond on my return.

Thanks and Regards

B

Hi Bobijohn,

the concept of generating L>1 wheels is the same as generating an open-cover or a simple close-cover (L=1) wheel. Again, we look for the least tickets required to offer the requested coverage. For example, an L=2 wheel simply means we guarantee to offer twice (L=2) the hits produced compared to the relevant L=1 wheel. In my previous example we have
C(13,5,3,4)=16 (L=1 wheel) and the respective C(13,5,3,4,L=2)=26 tickets. What you can see here is that we don't have to double the amount of tickets to provide twice the minimum guarantee which here is 2 #3-hits if we match 4 numbers.

The concept of numbers in groups looks in a wheel from a different perspective. That is, if we ensure e.g. we can match 2 out of 5 numbers, another 2 from another 5 numbers and so on, we ensure a particular guarantee. These types of wheels do not bother for the minimum lines required but are rewarding when their criteria are met.

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