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Simply put complex specified information is information that is both complex and specified, such that it is highly improbable and specific. The complexity of the information associated with event A is related to the number of bits I(A) associated with probability P(A) of a given event occurring such that I(A) = log_{2} P(A). The result is the more complex information is the more improbable it is.
One common example of complex specified information is credit
card numbers. Credit card numbers have 16 digests for a total of
10^{16} possible combinations. Now there are about 7 X 10^{9}
people in the world, if ever one had 10 credit cards that would
be 7 X 10^{10} active numbers. So probability of hitting
an active number is P = 7 X 10^{10}/ 10^{16}
= 7 X 10^{6} with
I = 17.124 bits. However the odds of getting a given
individuals credit card numbers would be P = 10/10^{16}
= 10^{15} with
I = 49.83 bits. So individual credit card numbers qualify as
complex information but because each number is associated with a
unique individual makes it complex specified information.
The best example of complex specified information is DNA. The DNA of each organism on Earth is unique, because of mutations and other factors, making it the most specified form of information known. The human genome contains more than 30,000 genes, at an estimated 3,000 base pairs per gene for a minimum of 90,000,000 base pairs or 90,000,000 base 4 bits. This results is 4^{90,000,000} or 10^{54,185,399} possible combinations the overwhelming majority of with are not viable. So the odds of hitting any individuals DNA by chance is P = 10^{54,185,399} with I = 179,999,999 bits. So DNA is both incredibly complex and specific.
Now let's see what means for a chance origin of life. First of all we need to estimate the number of planets where the conditions are right for life to get started. In this process will try to be has generous as possible.
Now based on the average estimated time that a star is on main sequence of 10^{10} years which is 3.156 X 10^{17} seconds and assuming one trial every nanosecond ( an extremely generous assumption) that comes 3.156 X 10^{26} per planet for a total of 3.156 X 10^{45} trials.
Now let us make another extremely generous assumption, that each trial is fully functional except for needing encoding all of the 124 proteins that even the simplest living organisms need to live or it dies and the process has to starts from scratch. These proteins have an average of 400 sequences amino acids. Since there are 124 of these proteins it requires a total of 49,600 sequences amino acids requiring 148,800 base pairs or 148,800 base 4 bits each. This produces 4^{148,800} or 3.36 X 10^{89586} possibilities. It needs to be noted that this is taking the simplest possible case as such reality would be a far bigger problem.
Assuming that the 124 proteins can be in any order there are 124! = 5.4 X 10^{205} possible successful combinations. So the odds of a successful trial is P = 5.4 X 10^{205}/ 3.36 X 10^{89,586} = 1.6 X 10^{89381}. Given a total of 3.156 X 10^{45} trials for the entire universe the odds of getting a successful trial is P = 1.6 X 10^{89381} X 3.156 X 10^{45} = 3 X 10^{89336}. There is a technical term in probability used to describe events with such small probabilities and that term is impossible. So it is statistically impossible to get proteins needed by even the simplest of living organisms.
The result is that information in DNA is so complex and specified that even making the most reasonable assumptions, it is impossible for the information in DNA to come about by chance.

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