Quoting an average success rate for rocket launches as a whole is not very useful. Atlas V has a near-perfect record, for instance, while Zenit and Proton have about 88% rates.
Presently, Falcon 9's partial success rate is 95% and its full success rate is 88% as an expendable, but it's still quite a young launcher. We won't know how it performs as a reusable until there's much more of a track record.
NASA has trusted not only satellites but humans to used rockets, many, many times. The space shuttles were used an average of 27 times each, and the two failures were not related to the reuse per se (although design decisions made in the name of reusability were major contributing factors). So it wouldn't be surprising for Falcon 9 first stages to manage 10 to 100 reuses, with some degree of engine overhaul or replacement.
If you mean provide rocket launch service at 1% of the per rocket lifetime costs, almost certainly not; that would be gambling on more than 100 successful reuses, a success rate which almost no expendable launch systems manage. It's much more likely that SpaceX will charge a large fraction of the price they would for an expendable rocket. Even charging half the current cost of a Falcon launch would be a win for the customers who currently pay full price and a win for SpaceX if they can use the first stage three or four times successfully.
Note also that Falcon 9 throws away its second stage, which is apparently around 25% of the total cost of the rocket (it accounts for half the guidance systems, half the tank assemblies, 1/7 of the dry mass, 1/10 of the engines, etc.) -- on that basis even a sixfold cost reduction would be a big challenge.