I'd suggest two things.
First, what you're offering is similar to an argument that Descartes offers elliptically in the Meditations. In Med. 1, Descartes points out that he has at times been mistaken. And as a consequence should doubt all of his beliefs, but that checking the beliefs would take an infinite amount of time. His solution is to instead suspend belief until he can come up with a firm foundation. Ostensibly in his argument, this is the cogito, but more accurately it's a circle of (a) the cogito (Med 2), his argument for a good God (Med 3), and "clear and distinct ideas" (scattered throughout and not well defined).
In the process, Descartes highlights a problem for the sort of system you're suggesting. Namely, there's going to be a negative infinite regress. In his case, it's that we have to keep doubting our judgments -- including our judgments about our judgments. In your case, it's a continuous loss of probability.
This leads to the second issue. Maybe both you and Descartes are wrong about what it means to know something? A lot of recent work has suggested that knowing is an act. This research is spear headed by Ernest Sosa and John Greco. On their view, to know something is more similar to successfully baking a cake than justified true belief. You either end up with cake or you don't.
On such a model whether you arrive at knowledge depends on the techniques you're using and virtues of the knower, and your confidence levels don't really enter in. Moreover, once the knowing is accomplished, it's over, so there's no room for the sort of compounding probably you and Descartes face.
Maybe to state it more simply, knowing may not be the sort of thing subjectable to an infinite set of regresses about our confidence in each act of knowing. Instead, it might just be something that succeeds or fails.