SOLUTION: A production process produces an item. On average, 15% of all items produced are defective. Each item is inspected before being shipped, and the inspector misclassifies an item

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Question 1205005: A production process produces an
item. On average, 15% of all items produced are
defective. Each item is inspected before being shipped,
and the inspector misclassifies an item 10% of the time.
What proportion of the items will be “classified as
good”? What is the probability that an item is defective
given that it was classified as good?
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
A production process produces an
item. On average, 15% of all items produced are
defective. Each item is inspected before being shipped,
and the inspector misclassifies an item 10% of the time.
What proportion of the items will be “classified as
good”?
 
P[(good AND cl. good) OR (def. and cl. good)] =(0.85)(0.90) + (0.15)(0.10) = 0.78

What is the probability that an item is defective given that it was classified
as good?
P(def.|cl. good) = P(def. AND cl. good)/P(cl. good)

P(def. AND cl. good) = (0.15)(0.10) = 0.015

P(cl. good) = 0.78 

P(def|cl. good) = P(def AND cl. good)/P(cl. good) = 0.015/0.78 = 15/780 = 1/52

Edwin
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Let's say 1000 items are produced. There's nothing particularly special about this value other than its some large round number. Feel free to pick something else if you want.

15% of those 1000 items are defective, so we have 0.15*1000 = 150 defective items and 1000-150 = 850 working items.

The inspector gets things wrong 10% of the time.
Of the 150 defective items, 10% of them are considered "good" when they shouldn't be. That's 0.10*150 = 15 items misclassified so far.
The remaining 150-15 = 135 defective items are properly labeled as such.

Of the 850 working items, 0.10*850 = 85 of them are considered "not good" even though they should be considered "good".
The remaining 850-85 = 765 working items are considered "good"

Here's a table summarizing everything
GoodNot goodTotal
Defective15135150
Not Defective76585850
Total7802201000

We have 780 items classified in the "good" column. So the proportion of good items is 780/1000 = 0.78 (i.e. 78% of the items are classified as "good").

Of that total in the "good" column, 15 are defective.

15/780 = 1/52 is the probability of selecting a defective item if we know 100% it was classified as "good".

1/52 = 0.01923 = 1.923% approximately
Since the decimal and percentage values are approximate, it might be best to stick to the fraction form.


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Summary

Question: What proportion of the items will be “classified as good”?
Answer: 0.78

Question: What is the probability that an item is defective given that it was classified as good?
Answer: 1/52
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