Question

Ponderosa paint & Glass makes paint at 3 plants. It then ships the unmarked cans to a central warehouse. Plant A supplies 50% of the paint & past records indicate that the pain is incorrectly mixed 10% of the time. Plant B contributes 30%, with a defective rate of 5%. Plant C supplies 20%, with painted mixed incorrectly 20% of the time. If Ponderosa guarantees it's product & spent $10,000 replacing improperly mixed paint last year, how should the cost be distributed among the 3 plants?

Answer 1

"the pain is incorrectly mixed"

Oww.

Answer 2

the paint is incorrectly mixed

Answer 3

Let \(D\) be the event of mix error.

Answer 4

sorry

Answer 5

Let \(A, B, C\) be the probabilities they came from each respective plant, ok?

Answer 6

Plant A supplies 50% of the paint \[
\Pr(A) =0.5
\]

Answer 7

incorrectly mixed 10% of the time\[
\Pr(D|A)=0.1
\]

Answer 8

This works for each plant the same way. Does this make sense?

Answer 9

okay. yes make sense

Answer 10

spent $10,000 replacing improperly mixed paint last year

This basically means when we sort out this money, we're assuming mixing errors happened.

Answer 11

So the amount given to plant A would be \[
\Pr(A|D)
\]

Answer 12

It works the same for the other plants by symmetry.

Answer 13

So your first step is to find out what \[
\Pr(D)
\]is.

Answer 14

Can you do it? It's the same as the last question.

Answer 15

okay i'll calculate it first

Answer 16

Did you get it?

Answer 17

Okay really? I have to go soon. Please hurry.

Answer 18

0.105 correct?

Answer 19

Yes.

Now do \[
\Pr(A|D)
\]

Answer 20

??

Answer 21

\[
\Pr(A|D) = \frac{\Pr(A\cap D)}{\Pr(D)}=\frac{\Pr(D|A)\times \Pr(A)}{\Pr(D)}
\]

Answer 22

You already calculated \(\Pr(D\cap A)=\Pr(A\cap D)\) when you tried to find \(\Pr(D)\)

Answer 23

http://www.wolframalpha.com/input/?i=(.5)(.1)%2B(.3)(.05)%2B(.2)(.2)

http://www.wolframalpha.com/input/?i=(.5)(.1)%2F(.105)

http://www.wolframalpha.com/input/?i=(.3)(.05)%2F(.105)

http://www.wolframalpha.com/input/?i=(.2)(.2)%2F(.105)

Answer 24

mm wait im confuse in P(A|D) given?

Answer 25

oh wait sorry my bet

Answer 26

thnks a lot!

Answer 27

do i multiple all the answer with 10 000? @wio

Answer 28

Yes.

Answer 29

Notice that the probabilities all add up to 1.
That is a good sign!

Answer 30

yupp thank you!