  An unconstrained nonlinear binary optimization problem of selecting a maximum
expected value subset of items is considered. Each item is associated with a
profit and probability. Each of the items succeeds or fails independently with
the given probabilities, and the profit is obtained in the event that all
selected items succeed. The objective is to select a subset that maximizes the
total value times the product of probabilities of the chosen items. The problem
is proven NP-hard by a nontrivial reduction from subset sum. Then we develop a
fully polynomial time approximation scheme (FPTAS) for this problem.
