We produce an industrial chemical product by processing batches of raw material. Historically, 80% of raw material batches have been "high quality", while 20% have been "low quality". Running a batch of low-quality material through our production process incurs an incremental cost of $3000, as compared to processing a high-quality batch. Unfortunately, it is impossible to tell for sure if a batch is high or low quality without processing it.
At an incremental cost of $1000, it is possible to "purify" a batch of material before processing it. Once the purification process is complete, the batch of material is certain to be of high quality.
Another option is to run a diagnostic test on a batch of material, at a cost of $100. The test is not perfect, but yields some potentially useful information, yielding an answer of "good" or "bad" on each batch tested. Based on past experimentation, you have determined the following:
Note that, unlike the previous problems of this kind we have encountered, the probabilities given here could all be directly estimated by real experimentation and data collection.