Question

A widget company tested 101,000 widgets, of which only 1,000 had defects. The company’s test correctly identified defects 99% of the time and correctly identified non-defects 95% of the time. 2 (a) What percentage of the widgets marked as defects were actually defective? (b) What percentage of the widgets marked as non-defects were not defective? (c) The answer to problem a is counter-intuitive. Explain what happened? (d) What would be your suggestion for improving the accuracy of the defect/nondefect identification. Improving the accuracy of the tests is not a valid answer.

Answer 1

Hello Sir/ Mam

Let D be the event that the widget is defective.

Let ND be the event that the widget is non-defective.

Let I be the event that the widget is found to be defective.

Let NI be the event that the widget is found to be non-defective.

Hence, given that:

Now, using total probability formula,

Hence,

(a) THE REQUIRED PROBABILITY = 16.53%

(b) THE PREQUIRED PROBABILITY = 99.9894%

(c) The answer to the problem a, as given above is intuitive as the company although tests the defects correctly 99% of the times, but as the total quantity of defective products is significantly low and hence, the probability of identification of a defective item(5.93%) is significantly higher than the actual probability of defective products(1/101). Hence, the result is intuitive.

(d) There is a huge lacking in the test where the company identifies the non-defective products as defective 5% of the times, which is a hige number. In those cases or while performing those tests, company can increase the benchmark level of identification and hence, rectify the identification of non-defective as non-defective by ignoring the non-defective products just at the sideline of the benchmark of being defective.

Sir/ Mam, in case of multiple questions asked in a single post, we are required just to answer the first. Hence, I'll request you to please ask the second one in a seperate post. Thanks!

I hope this solves your doubt.

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