Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among $n$ discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location $i$ takes $t_i$ time units and detects the hider -- if hidden there -- independently with probability $α_i$, for $i=1,\ldots,n$. The hider aims to maximize the expected time until detection, while the searcher aims to minimize it. We present an algorithm to compute an optimal strategy for each player. We demonstrate the algorithm's efficiency in a numerical study, in which we also study the characteristics of the optimal hiding strategy.