Almost always almost right
For any given objective function, let BotL (best-of-the-list) be the algorithm that evaluates the objective function on a set of inputs and returns an input from that list that had maximal output (from among those inputs), with ties broken arbitrarily. $\:$ Since APX only includes problems

who's objective function can be computed in deterministic polynomial time, BotL can be implemented deterministically in polynomial time. $\:$ Furthermore, the value returned by BotL

is at least as good as any of inputs in the least that BotL was evaluated on. $\:$ In particular,

if any of the inputs in that list is good-enough then BotL's output will be good-enough.

Therefore running BotL on the outputs of a sufficiently large number of independent executions of a base algorithm can amplify the success probability from 1/poly to 1-(1/(2^poly)).

As consequence of the preceding paragraph, the precise

confidence level essentially does not affect the resulting class.

(This situation is highly analogous to RP.)

I was not able to find anything about that on the complexity zoo, although there

might have been talks on it given at the workshop referred to in this paper.