True, I added a new element there that complicated the question. I'll rephrase.
What if someone in all seriousness says the "sinners prayer", truly intending to follow God, but immediately goes back to a life of depravity, never speaks of Christianity, never meets with other Christians, never reads scripture. Just said the words. Do you believe they are saved?
The question is whether a simple affirmation of belief is all that is required, or whether someone needs to actually repent, ie honestly attempt to change their ways, to be saved. Is salvation by grace alone and granted to the person who sincerely professes belief, or by grace alone but granted to the person who sincerely repents of their sins and professes belief. It is about the role of repentance in salvation.