"You can have a dozen children, but still the chances are that only half will have the mutant gene, and only half of that half would likely be passed on, and of the grandchildren of your children the chances are that only half of that quarter of your progeny would pass on that trait. - me- This is where the "selection" part comes in.
You are absolutely wrong. A child gets half his genes from the mother and half from the father. This is mostly a random chance process and it is the basis of genetics. This process does not "know" which genes are better than the other. Selection only operates after birth.
The children who get the mutation live. The children who don't, will die and not produce offspring. Genes can only be passed on by living people when they have kids.