Suppose an object is hidden in one of $N$ boxes. The initial prior probability that it is hidden in box $i$ is known to the searcher, who once a day must choose a box to be searched. The probability that the $j$th search of box $i$ will be unsuccessful, given that the object is in box $i$, is described by a random variable $\alpha_{ij}$. Assuming that each search of box $i$ costs $c_i > 0$, sufficient conditions on the joint distribution of the $\{\alpha_{ij}\}^\infty_{j=1}$ are found in order to guarantee that a particular search rule (analogous to earlier search rules found independently by R. Bellman, D. Blackwell, W. Black and J. Kadane) is optimal with respect to minimizing the total expected search cost of finding the object. An extension of a search problem studied by C. Sweat is also treated.