Probabilities in the Galaxy
A Distribution Model for habitable Planets
Copyright © Klaus Piontzik Claude Bärtels

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6.2 - Distribution of Civilization Levels

It can be assumed that the civilizations present in the galaxy will be distributed over the entire historical development evels. In a first approach one could assume that all civilizations are equally distributed over the levels of civilization.
On the other hand, it can be argued that the longer a civilization exists, the greater the probability of an all-destroying catastrophe. It is therefore more probable that the number of civilizations will decrease as the level of development increases. The following approach can be formulated from this:

6.2.1 Approach The probability Fz for a civilization is inversely proportional to the level of development.

F
z = 1:m and m = development level


This is illustrated once again in the following graphic.
probability Fz for a civilization
If intelligent life has been created on a planet, it is 100% likely that preliminary levels of civilizations have also been created, exactly those of level 1.
But already at level 2 the probability is only 0.5, at level 3 still 0.33, at level 4 only 0.25 etc.

It must now be demanded that the sum of all probabilities for the levels of civilization (m > 2) is equal to one. This means that it is 100% probable that at least one exists at the sum of all levels of civilization.
So the rule is:

6.2.2 Equation Sum of all civilization level

However, this is not the case with the previous approach 6.2.1, because totalling provides:

S = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 = 481 : 280 > 1

However, probabilities cannot become greater than one. We must therefore take a tougher approach.

A better approach can be achieved by using the square of the development level to calculate the probability F
z

Fz = 1:m2 and m > 1 is development level

The totals are provided here:

S = 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64
S = 7.301 : 14.400
S = 0.507,013

Then the normalization function 6.2.2 can be modified in this way:

6.2.3 Equation Sum of all civilization level


In order to adjust the developmental step function, the factor a must be equal to the reciprocal of the sum, thus:

a = 14,400 : 7,301
a = 1.972,332

Thus, the following approach can now be established for the probability of a development level:

6.2.4 Equation probability of a development level m > 1


The following graphic shows this once more.

Function probability of a development level

The development level function 6.2.4 can still determine the total probability for all civilization levels greater than 2.
The total probability is represented by the area spanned under the function. So you have to form the integral above the function. This can then be formulated as follows:

The total probability for the existence of a civilization corresponds to the integral over the probability function of a civilization.

6.2.5 Equation Integral over the probability function

Integral over the probability function

FAll = 14400/7301 · (-8-1 + 2-1)
Fall = 5400/7301 = 0.739,624

This means that there is a total probability of encountering a civilization at all of about 74%.

With the probability function for civilization levels, an effective tool is now available to describe and classify the probabilities for the development levels of a civilization. Now only one thing has to be taken for granted:

6.2.6 Axiom All considerations on levels of development, development time, and the distribution of levels of civilization are transferable to extraterrestrial civilizations.

 

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