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I'm not a biologist so bear with me. I know that the DNA molecule carries extremely large amounts of information. If the process of evolution is driven by completely random process .I think about it as the complex interactions between environment and random genetic mutations that leads to natural selection. I've two questions:

What's the probability of evolution of intelligent human beings within this scenario? I've heard that the probability of spontaneous generation of a good protein molecule should be nil in our universe: is this correct ?

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This question has a few problems at the moment, the two questions are pretty much creationist strawman arguments, which are not on-topic here. The probability of intelligent life evolving is not really answerable in my opinion, but the question about the evolution of a "good" protein (needs a better definition) would be okay. – Mad Scientist Feb 7 '13 at 15:41
good protein molecule = one that has biochemical activity, which is often one that has regular structure. some good work on the density of well structured protein sequences in sequence space has been done, but I don't have them myself. – shigeta Feb 7 '13 at 19:45
While some don't like Richard Dawkins, he did bother to write a book about the probability of life and how evolution has the ability to make improbable things reality. As such, I think this question is legitimate and often asked. If we can't explain fundamental biology to the public who needs such an explanation we are really limiting what the site should be used for in a detrimental way I feel. – shigeta Feb 7 '13 at 19:47
I think it could be improved as a question - for example, actually having a question in the title. – rg255 Feb 8 '13 at 13:11

If the process of evolution is driven by completely random process...

It's not. The evolution of "better" protein (and other) molecules happens because of selection, a very non-random process. The repeated selection of better molecules, and then of the variants of the better molecules, repeated many times, will lead to "good" molecules (in the sense that they work better for their producers).

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Exactly how genetic material changes from one generation to the next is a very complex subject. But essentially you are right. The change in genes from one generation to the next is not only mutations though. It is also mixing of the genes carried by the male and the genes carried by the female. The selection process is also very complex, and sexual selection is, for some species, a large part of it as well.

As for the probability of intelligent life emerging... Obviously, it's happened. The probability is at best extremely difficult to even provide an estimate of. How do you define a "good" protein molecule :-)?

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Since we only have one planet that we know of with life, it's a bit difficult to make good estimates on the probability of various events in the history of life. To make a good estimate, you'd want to have thousands of planets very similar to earth to compare. Since we don't have access to that kind of data, one proxy which you can look at is how long did it take for some event to happen in earth's history. In a vacuum you'd expect things that took longer to happen to be rarer. From this point of view, it seems likely that the development of multi-cellular life from simpler life forms is a low probability event because it took 3 billion years of simpler life forms before it happened.

Another way to get information on probabilities is to look at widely separated branches of the tree of life. Since wings evolved separately in birds, bats, insects, and pterosaurs, the probability of wings evolving in a planet with a similar atmosphere and multicellular animals is likely pretty high. Similarly, you can argue that the development of human level intelligence is relatively low probability because it has only developed once in the history of the planet in one lineage that very nearly died out. Dinosaurs never developed human level intelligence, nor did rodents, or mollusks, or arthropods. On the other hand, it seems that a lower level of intelligence at say the dog level is relatively high probability (since it has evolved separately in mollusks, birds, mammals, etc.).

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