Case context: You lead the team for a warehouse robot that scans labels and routes packages. The system has a 12% overall error rate in sorting items. An error analysis on misclassified cases shows that 8% of the mistakes come from reading barcodes incorrectly, while 30% come from missing damaged packages that should be removed from the line. One engineer argues that barcode reading should be fixed first because it looks simpler.

Question: Using the idea that the fraction of errors in a category sets a ceiling on how much total error can be reduced, which project should you prioritize? Compute the best possible overall error rate if you perfectly fix barcode reading, and then compute it again if you perfectly fix damaged-package detection.

Sample answer: You should prioritize damaged-package detection. The fraction of errors in a category limits the maximum total improvement from fixing it. If barcode reading accounts for 8% of all mistakes, eliminating that problem can reduce the overall error rate by at most 8% of 12%, which is 0.96 percentage points. The best possible overall error rate would be 11.04%. If damaged-package detection accounts for 30% of the mistakes, fixing it perfectly could reduce the overall error rate by 30% of 12%, or 3.6 percentage points, bringing the total error rate to 8.4%. Even if barcode reading seems easier, the damaged-package issue offers much larger potential improvement.

Key points:

• Identifies damaged-package detection as the higher-priority project.
• Applies the error-category-fraction ceiling correctly.
• Calculates the best possible error rate after fixing barcode reading (11.04%).
• Calculates the best possible error rate after fixing damaged-package detection (8.4%).

Rubric: The learner must choose damaged-package detection as the priority, explain that the fraction of errors in a category limits the maximum gain, and correctly compute both theoretical best overall error rates.