Thanks for the effort you've put into your reply. It's always appreciated! I agree that the chance of getting the number 7 out of a word count is much higher than the odds I was working with (although I did another calculation in an earlier reply based on your probability and it was still very unlikely, nothing like 1 in 8.6.) On the other hand, though, the chances of 2701 being a trefoil number are much less than the odds I worked with.

So, as it stands, the probability I worked out, which, after I found a powerful enough binomial calculator, was2 in a trillion, gives an initial ball-park estimate of the probability of finding 9 trefoil numbers out of ten numbers. You seem to be looking at the problem in a way that creates needless complexities (for a ball-park estimate, that is - I know you're aiming for accuracy here). If you look at the ten numbers purely as numbers you have 2, 7, 7, 28, 28, 73, 82, 298, 703, 2701. If we were to randomly pick 10 numbers between 1 and 3000 we would in most instances get higher numbers. The Genesis-derived numbers are therefore skewed towards the lower end, where the probabilities of hitting a trefoil number are higher, since the distance between members of each of the four sequences gradually increases. It's obvious that although the final probability will be nothing like 2 in a trillion, it will be high. Why? Because if the Genesis-derived numbers were not deliberately chosen, they would be blind regarding figurate properties. So the fact that all but one of them (2) are trefoil numbers and that 8 out of these 9 numbers make trefoil pairs, directly related to the prevailing G-triangle geometry within Genesis 1.1, is obviously of great significance if it can be shown to be improbable. Any set of numbers is equally improbable, but most of them are no little or no significance with regards to having related properties both to each other and to other numbers derived from the verse. That is where the significance of the Genesis-derived numbers lies.

Probability theorists will have ways of accounting for this skew towards lower numbers in the 10 values I 'picked' out of the bag of 3000 balls, but it seems obvious to me that you can obtain a very approximate answer to the question "What are the odds against 9 of 10 numbers randomly picked out of a bag of balls numbered 1 to 3000 (with replacement), happening to be trefoil numbers?" You can get an idea of the range of possible probabilities by taking the individual probabilities for the highest and lowest numbers in the set. The highest possible probability is based on the lowest value, 2. For the natural numbers up to 2 (1 and 2) there is a 50% probability of one of them being a trefoil number, 1 being a trefoil number. So if we assume there is a 50% probability of each of them being a trefoil the binomial probabilty of 9 out of 10 being a trefoil (equivalent to tossing a coin 10 times and getting 9 heads) is about 0.01, or 1 in 100. The lowest possible probability will be much lower than 2 in a trillion, maybe 1 in a quadrillion or thereabouts. It's obvious that the overall probability will be way below 1 in a 100, therefore, and easily below 1 in a million, I would guess. That's before we look at the extraordinary fact that the ten numbers are in pairs, the number from the verse being paired with the number from the last two words in the verse, and that four of the five pairs are trefoil numbers related to each other via G-triangle geometry! That itself is extraordinarily unlikely to have occurred by chance.

A separate calculation for each variable would be more accurate, and if I get time I'll do one. The separate probability would then have to be manipulated to get an overall average probability, but I'm not sure how to do that. Take the 10th root of the products of the 10 averages? I've already dipped my toes in the waters of probability theory and I'm not sure I'm capable of going further, not without a lot of training. Even then it might not be of much value, as some of the world's top statisticians and probability theorists disagree entirely about the reality of the Torah codes.

No, I'll stick to the kind of back-of-an-envelope maths I've always done. It has served me well and it is, I believe, good enough to give a very rough idea of how unlikely this phenomenon is. Remember, all I am calculating is the probability of 9 out of those 10 numbers having the properties of rhombic and triangular trefoils, based on numerical triangles and centred triangles. Yes, there is a relationshi between the number pairs, because a small number of letters, for example, must yield a small number of words. But there is zero relationship between any of the derived numbers and the properties of figurate numbers. It simply CANNOT be as high a probability as 1 in 8.6. It MUST be far lower than that. As a working estimate, I would go for 1 in a million, but I bet even that is conservative.

There are other types of trefoils, of course. You might like to see this one, which I derived from Vernon's Creation triangle.

Here we see the name and title of Jesus in Hebrew, Yehoshua HaMashiach, related to the Creation triangle, the Star of David and the threeness of trefoils.

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