On a Searching Problem
Олег Васильевич Староверов · Theory of Probability and Its Applications · 1963
Let the point with probability $p_k > 0,k = 1,2, \cdots ,n$, be located in a cell with the number $k;\sum _{k = 1}^n p_k = 1$. Only one cell is inspected per unit of time. If the point lies in the cell being inspectted, it can be discovered with a probability $p > 0$. The results of such investigations are independent. Let us denote by $\alpha _t ,1 \leqq \alpha _t \leqq n$, the number of the cell investigated at time t if the point was not discovered up to the time $t - 1$. Let $\alpha = (\alpha _1 ,\alpha _2 , \cdots ,\alpha _t , \cdots )$ be the procedure of searching and $\tau _\alpha $ the time required for discovering the point. In this paper a procedure of searching $\alpha ^ * $ is determined so that\[ {\bf M}\tau _{\alpha ^ * } = \mathop {\inf }\limits_\alpha {\bf M}\tau _a . \]