  We analyse the performance of well-known evolutionary algorithms (1+1)EA and
(1+$\lambda$)EA in the prior noise model, where in each fitness evaluation the
search point is altered before evaluation with probability $p$. We present
refined results for the expected optimisation time of the (1+1)EA and the
(1+$\lambda$)EA on the function LeadingOnes, where bits have to be optimised in
sequence. Previous work showed that the (1+1)EA on LeadingOnes runs in
polynomial expected time if $p = O((\log n)/n^2)$ and needs superpolynomial
expected time if $p = \omega((\log n)/n)$, leaving a huge gap for which no
results were known. We close this gap by showing that the expected optimisation
time is $\Theta(n^2) \cdot \exp(\Theta(\min\{pn^2, n\}))$ for all $p \le 1/2$,
allowing for the first time to locate the threshold between polynomial and
superpolynomial expected times at $p = \Theta((\log n)/n^2)$. Hence the (1+1)EA
on LeadingOnes is much more sensitive to noise than previously thought. We also
show that offspring populations of size $\lambda \ge 3.42\log n$ can
effectively deal with much higher noise than known before.
  Finally, we present an example of a rugged landscape where prior noise can
help to escape from local optima by blurring the landscape and allowing a hill
climber to see the underlying gradient. We prove that in this particular
setting noise can have a highly beneficial effect on performance.
