Interesting article with obviously a lot of work. My knowledge on this topic is almost zero but the direction of the thinking seems to be clear. Perhaps the conclusion is obvious but I felt it's not really clear. You ended your research with:

This is why the time is overdue to unmask this blind passenger, let him row back to his country of origin, the land of fantasy, and open the mind for the obvious origin of the immense and beautiful variety of all vegetal and animal species.

I just wondered, what is the obvious origin you're mentioning? Just curious.

Hello,

I finally solved all problems and finished my paper. Took a lot of time... I published it on academia.edu. If you are interested, you can see it here. Any comments and suggestions are highly appreciated. You can do this directly in the article.

Maybe your method is much better than mine for this kind of problem but it still lacks something as it does not yield the same result than my solution, from which I know that it is exact because it agrees with the numerical solution (see the plot below: blue points=numerical result; red line=analytical result):

What I got is the following for a=4, v=2 and v=3:

P[2] = -(3/256) m (-8 + 3 m);
P[3] = -((3 m (-128 + 80 m - 48 m^2 + 21 m^3))/4096);

Your solution does not seem to yield the same. Maybe this is because the letters are not independent from each other, because if a trial v is successful requires that all letters be correct, that is, correspond to the target. Also, whether a letter is correct or not on trial v depends on the issue on trial v-1, that is, on the previous one. For instance, if it was correct previously and does not mute it will be correct. This probability is: P(correct and does not mute) = 1/a (1-m). On the other hand: P(false and does mute to the correct number) = (1-1/a) m/a This is not the same. So there are conditional probabilities that possibly you did not consider. But your method may be improved. At the moment, I don't see clear since I am not familiar with Markov chains. But I will definitely take a closer look at this.

I am looking for an analytical solution for the problem Richard Dawkins has proposed numerically in his book "The Blind Watchmaker". His goal was to show that functional pieces of DNA can be formed by chance within reasonable time limits. In order to do this, he used a similar algorithm than mine shown in my first post. His target was the sentence from Shakespeare "methinks it is like a weasel". So a=27 here, that is, the length of the English alphabet with a space included. This sentence can be reached within only some steps if natural selection enters into the process. To simulate this, he created a parent in the first iteration, that is, an arbitrarily chosen series of 28 letters. Then he made kids of this parent by coping the parent a number of times. But these are not verbatim copies. Each letter can mutate with probability m when it is copied. Then he takes the kid that is the closest to the target and makes out of him the parent and the algorithm restarts. He showed that in this manner the target is reached after some 50 iterations or so. In my algorithm above, I did not include natural selection yet. Numerically, this is easy to do. But it is more difficult to get an analytical solution, which is what I am working on right now. I simplified the algorithm with numbers instead of letters (works faster) and it has a reduced alphabet. I found this probability distribution above with matrices in the case where the target is {2,5}. It doesn't matter what the target is. It could also be {1,1}. So my goal is to find an analytical probability distribution for a target of arbitrary length t: what is the probability P(v) that the target is attained in v trials. From this could then be calculated the mean value of v, which is a function of a, m, t and the number n of kids produced. With such a function, it would be possible to estimate whether natural selection is a probable process. This is the final goal.