Optimizing Defective Rates with Bayes' Formula

[CharlesDamle]

TL;DR Summary

Find least total defects of companies using Bayes formula.

Hello!
I'm sitting with a problem that is causing me some troubles..
First part is using Bayes formula.
We have 3 companies that produce some apparatus. Each company has some defective percentage.

Company	Produced (%)	Defective (%)
A	45	3
B	25	6
C	30	5

1) Suppose we pick up a defective apparatus. What is the probability it came from company A?
Here I use Bayes formula:
Defining the production in fractions.
Calculating the conditional probability (PD) and finding the probability it came from company A (PAD) to be 31%

2) If a defective apparatus is found, which company is most likely to have produced it?

Here I just calculate it for all three and find it is most likely company B and C

3) Now assume all the facilities have the same probability of being identified with a defect apparatus.
We want the least total defective apparatus and keep the production percentages the same.
How does the percentage of defects change for each company? The defective rates can only increase.

This is the one causing me problems.

The way I tried to solve it, was to setup 3 Bayes equations with 3 unknowns, namely the new defective rates.
They are all equal to the same, as they are asked to have the same probability -> P = 1/3
I end up with this, where x1, x2 and x3 represents the respective companies (A, B and C).

However this assumes that the defective rate of company A (x1) remains constant, and it says in the problem description it can only increase.
Am I even on the right track? I feel like I have to differentiate, but I'm now sure what.
I'd appreciate any advice or hints and sorry for the long post.

Cheers!

 

[PeroK]

I don't understand what part 3) is asking.

 

[CharlesDamle]

I don't understand what part 3) is asking.

As I understand, it wants me to minimize the defective rate overall while ensuring the companies have the same probability of being identified with a defect apparatus.
Don't know if that helps.

 

[Office_Shredder]

Here's my attempt at reading it. We want each facility to be equally likely to have made any given defective apparatus, i.e. change some stuff around so that the answer to 2 is that they are all equally likely. We want to keep the production percentages the same, so the only thing we can do is change the defective rate. We want the least total defective rate, so we don't want to just Jack them all up to be high numbers. All defective rates being 0 would work, but it also says we can only make defective rates go up.So my rough conclusion is we just need to increase A's defective rate while holding B and C fixed so that the answer to 2 is they are all equally likely.

 

[PeroK]

So my rough conclusion is we just need to increase A's defective rate while holding B and C fixed so that the answer to 2 is they are all equally likely.

That's my conclusion too. Silly question, if you ask me!

 

[Office_Shredder]

It's not that silly. If you are a plant manager at A, you might be measured on some metric of what fraction of the defects come from you vs another factory, and conclude you can spend slightly less money on quality control in exchange for a slightly higher defect rate, and need to figure out how much higher your detect rate can go. It's not well motivated in the question, but I can imagine actual people doing computations that look like this.

 

[CharlesDamle]

Thanks for your replies!

So my rough conclusion is we just need to increase A's defective rate while holding B and C fixed so that the answer to 2 is they are all equally likely.

I might misunderstand you, but it says the defective rates can only increase, so how can we keep B and C constant?

 

[Office_Shredder]

You are only allowed to increase them means you cannot make them smaller. I don't think there's a requirement that you have to increase them all, or else the solution will just be to increase them by the smallest epsilon amount possible since you are trying to minimize your total defective rate.

 

[CharlesDamle]

You are only allowed to increase them means you cannot make them smaller. I don't think there's a requirement that you have to increase them all, or else the solution will just be to increase them by the smallest epsilon amount possible since you are trying to minimize your total defective rate.

Alright, that was my conclusion as well. If they were all different, one would have to remain the same. So is my result in the last image correct? It does ensure that they are all equal to 1/3.

 

[PeroK]

It's not that silly. If you are a plant manager at A, you might be measured on some metric of what fraction of the defects come from you vs another factory, and conclude you can spend slightly less money on quality control in exchange for a slightly higher defect rate, and need to figure out how much higher your detect rate can go. It's not well motivated in the question, but I can imagine actual people doing computations that look like this.

That only makes sense if you include the cost figures. And, in any case, if the production percentages are different, why try to equalise the number of defects?

 

[PeroK]

Alright, that was my conclusion as well. If they were all different, one would have to remain the same. So is my result in the last image correct? It does ensure that they are all equal to 1/3.

It's not clear to me what your answer is. In any case, you should be able to do the calculation in your head.

 

[CharlesDamle]

It's not clear to me what your answer is. In any case, you should be able to do the calculation in your head.

I express the defection rates of B and C in terms of A's defection rate (x1).
From your explanations I think it's correct, as 2) gives the same answer when I plug it in.

 

[PeroK]

I express the defection rates of B and C in terms of A's defection rate (x1).
From your explanations I think it's correct, as 2) gives the same answer when I plug it in.

I think they expect a number for A like 4%, or whatever.

 

[CharlesDamle]

I think they expect a number for A like 4%, or whatever.

Oh right, so basically I will hand it in as
A_D = 1*0.03 = 3.0%
B_D = 1.8*0.03 = 5.4%
C_D = 1.5*0.03 = 4.5 %

 

[PeroK]

Oh right, so basically I will hand it in as
A_D = 1*0.03 = 3.0%
B_D = 1.8*0.03 = 5.4%
C_D = 1.5*0.03 = 4.5 %

You've reduced the defect percentage for B and C. Which you weren't supposed to do.

 

[CharlesDamle]

I have might been sitting with this problem for too long, if it something I should be able to do "in my head".
You wrote I should just change some stuff around in question 2), but that is also dependent on the conditional probability from question 1)?

 

[PeroK]

I have might been sitting with this problem for too long, if it something I should be able to do "in my head".
You wrote I should just change some stuff around in question 2), but that is also dependent on the conditional probability from question 1)?

You just want: $$45D_A = 25D_B = 30D_C$$ Where, as above, you can keep $D_B = 6$ and $D_C = 5$.

 

[CharlesDamle]

You just want: $$45D_A = 25D_B = 30D_C$$ Where, as above, you can keep $D_B = 6$ and $D_C = 5$.

Okay, I made it way too complicated in my head.
Thank you very much!