Posted on October 21, 2000.

In Reply to: Probable but not likely 1-2-3-4-5-6 posted by Jim Schutte on October 18, 2000.

: I am not a maths expert but I argued with my brother on a gut feeling that in a lotto 6 draw it is extremly unlikley that 1,2,3,4,5,6 will be drawn. He maintains that it is a random series of numbers which are as likely to be drawn as any other set. I think that you agree with me when you say that such numbers are extremely rare and will be left out of selections along with strings of odd or even numbers etc.

: Can you explain in laymans terms why this is so?

: Can you do a page on your site on the basics of probability. You mention guassian curves and the law of big numbers. What are these? without being to technical of course.

: Great site

: Regards

: Jim

Jim:

It would take a lot more than a Web page to tackle theory of probability. Online probability tutoring would require a full-time job, and then some. At this time, I can only offer so much on my website. But what I offer is undeniable mathematically. I don’t try to fool anybody simply because I don’t fool myself. Normally, there are some who do not agree with my ideas. I am not saying that I am beyond error. I am saying, however, that those who argue over my theories do so based on their personal feelings, not on cold facts. I visualize Truth as a meteor wondering in the cold outer space. Regardless if humans like it or not, the meteor can still collide with the Earth. It all depends on what rule-based trajectories the two bodies follow…

Your observation on the “1-2-3-4-5-6” type of lotto combinations is also on many people’s minds. Indeed, every lotto combination has the same individual probability: p = 1/N. There are 13983816 lotto 6/49 combinations, therefore any combination, including 1-2-3-4-5-6, has a 1 in 13983816 chance to hit. That’s one step of the theory, the first one. There are a lot more steps. We can look at simpler lottery games, where the total of possible combinations is much lower, therefore easier to analyze. For example, the pick-3 game has only 1000 outcomes, from 000 to 999. The individual probability is 1/1000. One should expect one particular combination to appear every 1000 drawings. Further, since p is constant, all pick-3 combinations should appear equally (in multiples of 1000 drawings). Thus, 2000 drawings, every combination should have come out 2 times, etc. Does it ever happen? NEVER, EVER. There are combinations that come out 10 times in 5000 drawings, while other combinations do not appear at all.

The fundamental parameter that explains all this best: Standard deviation rules!

Briefly, there is more to lottery than just the individual probability. Look at many lotto drawings, thousands if possible. You can easily calculate the average of a lotto game. A lotto 49 game has an average of 1+49=50/2=25. You can use Gauss’s formula and notice something. The sum-totals of most drawings fall within the Gauss (bell) curve. It’s hard to find a 1-2-3-4-5-6 combination anywhere, or a 44-45-46-47-48-49 combination. There is a much more complex explanation for the absence of such combinations. It can use Markov’s decision-making theory (Markov chains). The easiest path in a Markov chain is totally random. A combination like 1-2-3-4-5-6 would require additional steps, which dramatically decrease the probability. Simply put it, the process will follow such steps as: “The 1st number is 1; is the 2nd number = 1st number + 1? If not, repeat the step; and so one until 2nd number = 1st number + 1; etc with all 6 numbers”.

There is another element: the fascinating median. The Romans had the saying “Aurea mediocritas” = “the golden middle”. The Ancients used also the “golden number” on a huge scale in their arts. The golden number is related to the median. The golden number is found all over the universe, such as in the shapes of the galaxies. The golden number is also an infinite series of 1 arranged in fractions. I don’t have a good equation editor to display here such sequence. It goes like 1 + 1 (over 1+ ((over 1+ (over 1 +)))… and so one to the infinite. Fascinating! There is no absolute 0 and there is no absolute 1 in the Universe. It’s all a continuous interaction between 0 and 1.

But this is too much for a web page…

In more down-to-earth terms, the median plays an important role why lottery combinations are never equal in frequency. I presented in several places on this website the role of the median. In a nutshell, every lottery number will repeat, in at least 50% of the cases, after a number of drawings less than or equal to the median.

- Theory of Probability: Best introduction, formulae, algorithms, software.
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Of interest:

- Fundamental Formula of Gambling (FFG).
- Mathematics of Fundamental Formula of Gambling.
- Download Probability, Mathematics, Statistics, Combinatorics
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