My son told me about an experiment they had in biology classes to introduce the dominance in traits and evolution.
- you start with 6 marbles of 6 different colors
- for each color, you randomly pick a number $n$ between 1 and 6 and add to an empty bowl $n$ marbles of that color (you have a large reserve of marbles of all the colors)
- you end up with 6 to 36 marbles of various colors in the bowl
- you then randomly pick 6 marbles from the bowl, empty the bowl, and this is the starting point of the next iteration (second bullet above)
Note: each of the 6 marbles is independent and the number which is picked for each of them adds that number of marbles of that color to the bowl. This means that if you have (I will just use three marbles here, the experiment is on 6 of them) red red yellow, you would pick (and put in the bowl) $n=\textrm{rand}(6)$ red marbles, then $k=\textrm{rand}(6)$ red marbles, and then $r=\textrm{rand}(6)$ yellow ones, ending with $n+k+r$ marbles in the bowl.
The conclusion after several experiments was that after about 7 generations you end up with 6 marbles of one single color. I believe the idea of the experiment is to show that a random process can lead to the preference of one trait over the others (but the interpretation does not matter for the question here)
He was interested in writing some code to simulate that, so I did my homework and wrote it in advance so that I can be the super dad who knows the answers to everything. I ended up with the following distribution of the number of generations needed for a set of six marbles of the same color after 100,000 iterations:
My concerns/questions are the following:
- is there a certainty that the experiment is convergent = that the iterations with the 6 marbles eventually finishes with a set of 6 marbles of one color? (intuitively and with some hand waving I think so)
- is there a way to quantify the distribution above, that it to show by calculation that for instance the average number of generations for 6 marbles will be 5.58 (this is an example - based on what I see in the distribution)
- or alternatively: is there a function (distribution) which corresponds to the experiment? (from which the average and other interesting numbers can be derived)