Chapter 4, Problem 4.29P

There are two possible causes for a breakdown of a machine. To check the first possibility would cost $C_1$ dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of $R_1$ dollars. Similarly, there are costs $C_2$ and $R_2$ associated with the second possibility. Let p and $1-p$ denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on $p,C_i,R_i,i=1,2,$ should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?

Note: If the first check is negative, we must still check the other possibility.