  "Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
  Amir, Beigel, and Gasarch (1990) show that for constant $k$, if $k$ instances
of a decision problem can be efficiently solved using less than $k$ bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
$\ell$ help bits, $k$ instances of the problem can be efficiently solved with
probability greater than $2^{\ell-k}$. The same result holds if using less than
$k(1 - h(\alpha))$ help bits (where $h(\cdot)$ is the binary entropy function),
we can efficiently solve $(1-\alpha)$ fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic $k$. In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "$k$-membership comparability," a notion
known to be related to solving $k$ instances using less than $k$ bits of help.
  Next we consider the setting of average-case complexity: Assume that we can
solve $k$ instances of a decision problem using some help bits whose entropy is
less than $k$ when the $k$ instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than $1/2$.
  Finally, we show that in the case where $k$ is super-logarithmic, assuming
$k$-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof."
