It is true that the probability is that you will have brown-eyed children. If there is  definite evidence that you do not carry the blue-eyed gene, then it is correspondingly sure that you will have no blue-eyed children at all (except that many brown-eyed children are blue-eyed at birth, only changing their eye colour duriong their early babyhood.) If you do in fact carry the blue-eyed gene, then the simplest expectation is that half your children would be blue-eyed, and that all your brown-eyed children would also be "carriers".

It has nothing to do with the concept of one gene being "stronger" than the other. You generally have two genes that affect that particular aspect of eye colour. Each leads to certain conditions, such as the production of a protein that can build up into structures that show blue or brown as the case might be. If the brown say, "masks" (hides) the blue colour, then although both structures are present, the eyes will look brown. In genetics we say that the brown gene then would be "dominant" to the blue gene, which then is the "recessive" gene. (This is just an example of how the dominance could work; there are far more subtle ways in various genes.)

If there is selection against just one of two genes for a particular trait such as eye colour, then in the fulness of time (quite quickly if the gene being eliminated is the dominant gene, only very slowly if it is the recessive gene) one might expect that gene to vanish, but otherwise the mix of genes might last in the population indefinitely.