Question

In: Statistics and Probability

Your production line has an automatic scanner to detect defects. In recent production, 3% of items...

Your production line has an automatic scanner to detect defects. In recent production, 3% of items have been defective. Given that an item is defective, the scanner has an 90% chance of identifying it as defective. Of the non-defective items, the scanner has a 95% chance of identifying it correctly as non-defective.

Given that the scanner identifies a part as defective, find the conditional probability that the part is truly defective.

Solutions

Expert Solution

Problem statement:- An operational process of detecting defective products is described. Based on this we have to answer the question asked.

Given:- A production line has an automatic scanner to detect defective products. Observed defective percentage rates is mentioned in the question which is 3%. Based on the information, we have to answer the question asked.

Solution:- To compute the conditional probability that the part is truly defective, let us assume there were 1000 product samples that were passed through the scanner.

Then,

The number of defective products= 3%(Given)*1000=30

The number of non-defective products= 97%*1000=970

In the information provided, it is given that 90% of the times, scanner detects a defective product when it is actually defective. For the sample considered, we get= 90%* number of defective products=27

In the information provided, it is given that 95% of the times, scanner detects a non-defective product when it is actually non-defective. For the sample considered, we get= 95%* number of non-defective products=921.5

Constructing the table by using the above computations we get,

Actual Product condition Product condition according to scanner
Non-Defective Defective
Non-Defective 921.5 48.5
Defective 3 27

Now to compute the conditional probability that the part is truly defective when scanner has identified as defective, we use the formula,

=(Number of actual defectives among the number of products highlighted by scanner as defective) / (Total number of products identified by scanner as defective)

=27/(48.5+27)=0.3576

conditional probability that the part is truly defective is 0.3576.

orchestra answered 2 years ago
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