what is the probability that part is actually defective? I've this "simple" yes it's simple, question, from which I got different outcomes, and I would like to have a differente opinion.
A manufacturing process has a 3% defect rate. Inspectors catch 95% of defects but also fail 5% of non-defective parts. If we pick a part at random from all those that pass inspection, what is the probability that part is actually defective?
On my side , I get 0,5% of probability that part is defective if picked randomly.
In the other hand, 0,16% seems to be the write answer.
If someone could share his result, It would be very much appreciated.
Rgds.
 A: This problem can be tackled using conditional probability.
We are seeking the probability that a part is defective, given that it has passed inspection. $Pr(\text{defective}|\text{passed inspection})$.
By Bayes’ theorem, this is equal to $$\frac{Pr(\text{passed inspection}|\text{defective})Pr(\text{defective})}{Pr(\text{passed inspection})}$$
We are given that $Pr(\text{defective}) = 0.03$ and we know that $Pr(\text{passed inspection}|\text{defective}) = 1 - 0.95 = 0.05$ because this is the probability that a defective item is not detected.
We can then calculate $Pr(\text{passed inspection})$ as
$$Pr(\text{passed inspection}|\text{defective})Pr(\text{defective}) + Pr(\text{passed inspection}|\text{not defective})Pr(\text{not defective})$$
(because there are two ways of passing inspection, one if the item is defective, and another if the item is not defective)
which is
$$0.05 \times 0.03 + (1-0.05) \times (1 -0.03) = 0.923$$
So for $Pr(\text{defective}|\text{passed inspection})$ we have $\frac{0.03 \times 0.05}{0.923} \approx 0.0016 = 0.16\%$.
A: Yes, I'll do that, I'll mark as accepted. 
Nevertheless, I still have something around my mind about this.
If we're given :
Defect rate 0,03 mean 0,07 of non defect right?
We can put this way:
O,03 defect
0,07 Non Defect 
When inspector catch 95% of defect  we can put that this way 
0,03  Defect    
**** O,05 Fail Inspect
**** 0,95 Succeed Inspec

while 
0,07 still non Defect right?
Still, 5% of Non-Defective are not Catch so we can represent like this
0,03  Defect    
**** O,05 Fail Inspect
**** 0,95 Succeed Inspect
0,07 Non Defect
**** 0,05 Fail inspect
**** 0,95 Succeed Inspect

So, the total probability of parts with defects if random select is = (0,05)x(0,03) + (0,05)x(0,07) = 0,15 + 0,35 = 0,50 
Now if I divide 0,5 by 0,03 (which is the percentage of defects parts) I get 0,16%
exactly the same value as yours, but this 0,16% in my interpretation is the  proportion of defective parts given Defectives. In other words, is the probability of pick a part selected random from the Defect only, and not from all inspects. 
I know, the odds are against me.
Thanks in advance for you feedback
