How can an intelligent person actually believe that he can prove God does not exist?

I cannot account for or explain my own existence, but my parents can, and my grandparents my parents' existence, and my great-grandparents my grandparents' existence, etc. If I carry this regress back far enough, I arrive at a first set of human parents, who cannot account for or explain their own existence, who must have been brought into existence by a being who is self-explanatory (and also infinite, eternal, independent, self-existent, and all the other traditional attributes of deity). In philosophical terms, the existence of contingent beings implies the existence of a necessary beings, one of the classical and traditional theistic proofs. - George T. Thompson

My response:
Well, you are however slightly different from your parents. Your parents are slightly different from their parents. We are evolving. To say that by regression, the first pair of human must be human is wrong. Homo sapiens share common ancestor with other hominds, great apes etc. which in term share common ancestor with othr animals and plants and so on.

In philosophical terms, one cannot prove non-existence of any object. It is up to those who claim god exists to proof that god exists. But here are a few simple steps to show the unlikelihood of god's existence.

1. Assume God created the universe.
2. God either created the universe and let it run its course, or intervene when necessary.
3a. A non-intervening god is what Einstein called a "cosmic god". It is not the personal god commonly referred to in most religion. Such a non-intervening god, if exists, does not make any difference to physical laws and our daily activities. By Ockham razor, we can ignore the existence of god.
3b. God is intervening. That means god answers prayers.
4. Prayers do not work.
5. Hence God does not exist.