Is bounded-error probabilistic computation sensitive to transition types?
If you have a coin which has an uncomputable probability of landing heads, then you can estimate with bounded error the first $k$ bits of this probability using $O(4^{k})$ coin flips. 

This lets you construct a machine with bounded error that computes an uncomputable function. The uncomputable function is the $k$th bit of the probability. The computation is estimating the probability of landing heads by flipping the coin $4^k$ times. Considerations of running time don't matter when you're worrying about whether a function is computable or uncomputable, so it doesn't matter that it takes time exponential in $k$ to estimate the $k$th bit of the function; it's still an uncomputable function.