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Complex specified information
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 the more complex information is the more improbable it is.
Common Examples
One common example of complex specified information is a credit card number. Credit card numbers have 16 digits, giving a total of 10^{16} possible permutations. Now there are about 7 X 10^{9} people in the world; if everyone had 10 credit cards that would be 7 X 10^{10} active numbers. So the 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 individual's 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 the fact that 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 contain 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 individual's DNA by chance is P = 10^{54,185,399} with I = 179,999,999 bits. So DNA is both incredibly complex and specific.
Now lets see what this 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 as generous as possible.
 There are an estimated 10^{11} stars in an average galaxy and an estimated 10^{11} galaxies, giving a total of 10^{22} stars.
 Current evidence indicates that about 1/2 of all stars have planets resulting in 5 X 10^{21} planetary systems
 Current evidence indicates that about 90% of planetary systems have gas giants in orbits that eliminate the possibility of terrestrial planets in the habitable zone of the star, so the maximum number of planetary systems with terrestrial planets is 5 X 10^{20}.
 Let be generous and assume that 10% of these system actually have terrestrial planets in the habitable zone with an average of 2 per system resulting in 10^{20} terrestrial planets in habitable zones.
 Let be generous again and assume that 10% of these planets have ideal conditions for life to form, this results in 10^{19} potentially habitable planets.
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 works out at 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; if a trial lacks these, it fails and the process has to start from scratch. These proteins have an average of 400 sequences amino acids. Since there are 124 of them, it requires a total of 49,600 sequences of 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 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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