count my change.

(In progress.)

There exist a number of ways in which to go about approximating the impact of any action taken upon a space. One might simply guess, or intuit. One might calculate the curvature of a set of small assumptions and extrapolate based off of what arises from there. This is my preferred method. Indeed, when all we have to work with are ranges and approximations, this is often the way, as we discover through the course of an intermediate mathematical education.

The accuracy of a conclusion based on several uncertain assumptions depends on the accuracy of each assumption - but the overall confidence shrinks with each additionl imperfect assumption. If I am 95% sure that I am right about some assumption I have made, that may seem fine. In many cases, it is. The danger lies in stacking assumptions. 95% can be represented decimally as 0.95.

How to get wronger.

0.95 x 0.95 = ?

95 x 95 = ?

5 x 5 = 25

5 x 90 = 450 (x2) = 900 (as I examine this, I notice that it would have been more efficient to have doubled the five from outset, but I pursue mathematics via spiral - 10 x 90; this line will remain here to demonstrate my self-revision process - 10 x 90 = 900)

90 x 90 = 8100 (9 x 9 = 81; 10 x 10 = 100; 100 x 81 = 8100)

95 x 95 = 9025

0.95 x 0.95 = 0.9025

Now that we've made two assumptions in a row, each with a generous 95% accuracy rate, we are probably around 90% right, by my angular math.

Do you notice how that is a multiple of nine, and some change? 9 x 1000 = 9000; 5 x 5 = 25. 9025. That pattern holds. 0.95^4 = 0.95 x 0.95 x 0.95 x 0.95 = 0.9025 x 0.9025 = 0.81450625. Make note that a wily enough datamancer could make this look like that long decimal, or like 0.81, or - by some stretches of the imagination, 0.82. This will be an important idea later. For now - hold in mind that even in the relatively objective world of numbers, interpretation often matters. And recall the truth of the multiples of nine - that they always add up to nine.

Suppose that, rather than begin with a 95% accuracy strength rating for each of our assumptions, we begin at 75%. Our approximation now looks as follows:

0.75 x 0.75 = ?

75 x 75 = ?

5 x 5 = 25 (Stack those easy wins.)

70 x 70 = 4900

2x(70 x 5) = 10 x 70 = 700

Add those totals (4900 + 700 + 25), and we arrive at 0.5625, or 56.25% accuracy rate, following two assumptions.

Note the trends at play. In both instances, we see progressively worse results over time. If we are exceptionally clever, our best-case scenario is that, as we make successive correct guesses, the likelihood of our continuing to guess correctly decreases at a slightly less depressing rate than the person who was guessing with an average success rate.

Oof.

How to right wrong, angles.

Consider the graceful hand of an artist.

That artist sat, incorrectly - if reasonably - drawing their own hand, for however long it took them. When we leave a stray line in our drawing, or brushstroke in our painting, we have a decision to make -

We can leave it:

How change matters.

Say that I've got one tree. That tree is somewhere between zero and fifty years old. It is an oak tree (or an elm, or a willow, or a pine tree). We are in an urban (or a suburban or a rural) domain. This results in our tree's having some Base Trunk Width (BTW). We are going to make a set of assumptions which arrive us at our Base Trunk Width (BTW).

From there (BTW), we are going to derive some nominal value, using as few speculative generosities as possible - an effort to remain in balance with Occam's Razor. We may need to make some assumptions and speculations together, but these will be rooted in reality, and their leaps small. This will generate our Expected Nominal Tree (ENT) value.