Question 1181356: Ace Machine Works estimates that the probability its lather tool is properly adjusted is 0.8. When the lathe is properly adjusted, there is a 0.9 probability that parts produced pass inspection. If the lathe is out of adjustment, however, the probability of a good part being produced is only 0.2. A part randomly chosen is inspected and found it to be acceptable. At this point, what is the posterior probability that the lathe tool is properly adjusted?
Please provide steps to solve this problem. Thank you
Answer by ikleyn(52788)   (Show Source):
You can put this solution on YOUR website! .
Ace Machine Works estimates that the probability its lather tool is properly adjusted is 0.8.
When the lathe is properly adjusted, there is a 0.9 probability that parts produced pass inspection.
If the lathe is out of adjustment, however, the probability of a good part being produced is only 0.2.
A part randomly chosen is inspected and found it to be acceptable. At this point,
what is the posterior probability that the lathe tool is properly adjusted?
Please provide steps to solve this problem. Thank you
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In this problem, they want you find the CONDITIONAL probability,
although they do not pronounce this term.
They want you guess it - - - the correct guessing
is a PART of the problem's solution (and is its MAJOR part).
So they want you find the conditional probability
P(lathe tool is properly adjusted given the randomly chosen part is acceptable).
In more formal way, they want you find
P(lathe tool is properly adjusted | the randomly chosen part is acceptable) =
= P(lathe tool is properly adjusted AND the randomly chosen part is acceptable) / P(the randomly chosen part is acceptable).
Regarding the numerator of this fraction, it is equal to
P(lathe tool is properly adjusted AND the randomly chosen part is acceptable) = 0.8*0.9 = 0.72.
Regarding the denominator, it is equal to
P(the randomly chosen part is acceptable) = 0.8*0.9 + (1-0.8)*0.2 = 0.76.
THEREFORE, combining these circumstances, we find the answer
P(lathe tool is properly adjusted given the randomly chosen part is acceptable) =  = use your calculator = 0.9474 (rounded).
ANSWER. The posterior probability that the lathe tool is properly adjusted is equal to 0.9474, or 94.74%.
Solved, answered and carefully explained.
And you have now the mantra in your archive to pronounce it every time as you solve similar problem.
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On conditional probability, see the lessons
- Conditional probability problems
- Conditional probability problems REVISITED
in this site.
Read them to develop your skills and knowledge.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Solved problems on Probability"
and "Additional problems on Probability".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.
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