# Probability of selling a bad part

Good Morning

We have  a part that is designed with an estimated 10^-6 failure rate.  We perform 100% inspection which is approximately 80% effective.

What is the probability that we will sell a bad part?

Thank You
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Commented:
How many parts do you sell?
Does 80% effective mean a false positive rate of 0.20 and a false negative rate of 0.20?
Author Commented:

Thank you
Commented:
Are you actually selling parts, or is this a homework question?

If you didn't do any testing at all, what would be the probability of selling a defective part?
Author Commented:
It is not a homework question, I am trying to confirm my own calculations.
Software DeveloperCommented:
I don't know, if I get the terminology of this right, but isn't it simply  the product of probabilities?

1 in a million parts is really faulty. All parts are checked you would detect 200.000 of a million as faulty, though they re not, but the 1 really faulty part would have a 20% chance to pass OK.

The chance for it to occur at all must be translated from the failure rate as 0.0001% (1% is 1 in 100,  .1% is 1 in 1000 etc.) multiplied by 20% means 0.00002% or 2 in 10 million parts. The other way around is saying 20% false positives mean 1 in 5 faulty get through and so the expected number of parts going through as OK, though being faulty is 1 in 5 million or 2 in 10 million, or 0.01 in a year of production. Formulated in another way your chances are 1% to sell a single faulty part in a years production, mainly because of the low failure rate.

Does that match your own conclusion?

The bad part about this setup is, that you throw away 20% of OK parts. So raising the fault detection exactness will reduce production costs, or doing no QA you still have a low rate of sold failures. As soon as a failure doesn't harm someones life it sure is easier to handle this via insurance or refunds than doing any QA at all, the QA is killing nearly 20% of the production without the real need. The whole thing gets more realistic with comparable failure rates and misdetection rates.

Bye, Olaf.
Author Commented:
Olaf

As you know, 100% inspection is 80% effective, I believe Deming came up with that one, but I don't have a direct reference at the moment.  It isn't that we are throwing away 20% of OK parts, 20% of the parts aren't adequately inspected just as an assumption taking Deming's observation into consideration.

As ozo observed above, with the false positive and negative rates inquiry, I am attempting this problem using Bayes' Theorem and wanted to see if my calculations were correct, and even if using Bayes' was necessary but didn't want to poison the well.  I wade in and out of this subject once every 5 years and thought I would give EE a try.

Commented:
If 100% inspection is 80% effective is really part of the process, you have to define it better.
If it is an irrelevant quote from Deming, you should have said so.
How good is your estimate for estimated 10^-6 failure rate?
Do you actually have data that is worth using for calculations?

If the failure rate is really 1 in 10^6, the probability of selling a defective part is also 1 in 10^6.
(Assuming failure and defective mean the same thing.)

If inspection catches 80% of the defective parts (4 out of 5), then the probability of selling a defective parts goes up to 1 in 5x10^6.

If you sell (and continue to sell) 100K parts per year, you can expect to sell one defective part every 50 years.
Commented:
"If the failure rate is really 1 in 10^6, the probability of selling a defective part is also 1 in 10^6.
(Assuming failure and defective mean the same thing.)

If inspection catches 80% of the defective parts (4 out of 5), then the probability of selling a defective parts goes up to 1 in 5x10^6.

If you sell (and continue to sell) 100K parts per year, you can expect to sell one defective part every 50 years. "

dglitch has illuminated your problem above, especially the last sentence which puts everything in a correct perspective.
the big unanswered information lies in the relationship between the estimated failure rate and the testing as has been pointed out.
Unless lift is involved, sell you parts with a replacement guarantee. If one fails your company may already been bought by somebody and it will not be your problem. Or the testing may cost more than the replacement.
In general there is no relationship between failure rate and testing effectiveness.
Commented:
"life" not "lift" in last paragraph
\Commented:
>> If inspection catches 80% of the defective parts (4 out of 5), then the probability of selling a defective parts goes up to 1 in 5x10^6.

i.e., probability of selling a defective parts = 0.00002 %
which is what Olaf said in http:#a41269705 : "means 0.00002% or 2 in 10 million parts"
"... so the expected number of parts going through as OK, though being faulty is 1 in 5 million or 2 in 10 million"
so not sure what different conclusion you are making.

But Olaf's alternate formulation looks incorrect:
"Formulated in another way your chances are 1% to sell a single faulty part in a years production, mainly because of the low failure rate."
I see where Olaf got the 1% (i.e., from his .01 calculation), but the .01 was not a probability.

@aasikolo,
Since two posts have come to the same conclusion, why don't you critique their results explaining why you believe it is wrong.
Ok, show your results and why your solution is correct and their solution is wrong.

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\Commented:
I just saw aburr's comment.
Or the testing may cost more than the replacement.

Isn't this saying the same thing as Olaf said in http:#a41269705
doing no QA you still have a low rate of sold failures. As soon as (sic. asuming "assuming that") a failure doesn't harm someones life it sure is easier to handle this via insurance or refunds than doing any QA at all
Author Commented:
Phorrific:

I didn't indicate that anyone's conclusions were wrong.  My statement was that my conclusion did not match their results.  My results are attached for your review and comment.  From ozo's observation, only the true and false positives were considered in my results, the true and false negatives were not.  Hence my results may be incorrect. (3.9 x 10^-6)

Failure-Probability.xlsx

Aburr:

As you know, failure rate and failure probability are not the same thing. Bayes needs the rate at which the inspection program in this case is effective.  Deming's 80% is an assumption based on the known effectiveness of 100% inspection programs, I was looking for worst case.

Thank you
Commented:
As far as I know, failure rate and failure probability are the same thing.

Deming's 80% is a quip to suggest that you should put more effort in quality production than into inspection.  I can't imagine any real inspection process for a precision part that would be only 80% effective.

I don't believe you are applying Bayes' Theorem correctly.
Commented:
My apologies.  After a head-clearing walk . . .

If you manufacture 5,000,000 parts
4,999,995 are good
5 are defective

After testing
4,999,995 x 0.8 of the good parts pass   ==> 3,999,996 to sell
but   5 x 0.2 of the bad parts do too  ==>         1 gets sold
Probability of selling a bad part is 1/3,999,997  approx 1 in 4 million.
Commented:
If you sell 100,000 per year,
then the chance that none of them are defective is (1-1/3,999,997)^100,000
so the chance of selling a defective part during the year is (1-1/3,999,997)^100,000 = 0.02469
The chance of selling a bad part sometime during n years would be 1- (1-1/3,999,997)^(n*100,000)
Software DeveloperCommented:
Overall it seems to me you're mixing rough estimates like the 80:20 rule with a precise calculation of probabilities via Bayes Theorem. It's quite useless to put these assumed probabilities into Bayes Theorem to get some unprecise prediction.

If you don't have 20% parts failing QA tests, then the QA precision is higher than 80:20 so the outset is wrong and you can't determine something with a precise calculation based on wrong estimates. If you don't have a rate of 80% false negatives which you don't throw away or reenter into the production process, then you already know the 80:20 estimate is wrong.

Bye, Olaf.
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