Question

The board of directors of a company knows that the probability that the carbon emission from the company’s manufacturing unit is exceeding the permissible level is 35%. A consultant is hired to use a carbon footprint calculator to test for the emission level. The accuracy of this test is said to be 85%.

You have this additional information: the test reported that the carbon emission from the manufacturing unit is well within the permissible level. What is the probability that the emission is within the permissible level, given the outcome of the test?

Answer 1

0.0135
0.2238
0.3363
0.6041
0.9132

Answer 2

@YanaSidlinskiy

Answer 3

@TuringTest yana told me to ask you for help.

Answer 4

I would start by making a tree diagram of the situation.
Let A=the carbon emission from the company’s manufacturing unit is exceeding the permissible level
Let B=The test is accurate

We then want \(P(A|B)\), do you agree?

Answer 5

That is correct/

Answer 6

I would start by making a tree diagram of the situation.
Let A=the carbon emission from the company’s manufacturing unit is exceeding the permissible level
Let B=The test is accurate

We then want \(P(A^c|B)\), do you agree?

Answer 7

I had A instead of A^c, sorry

Answer 8

I did not even notice that.

Answer 9

I am going to be completely honest I have no clue how to do this problem. Can we go through it step by step?

Answer 10

so the tree diagram would beI feel like I may be misinterpreting what they are asking for, let me think for a sec :P

Answer 11

ok we need one more thing, sorry
We have the probability that the maximum level is exceeded is A, and that the test is accurate is B
Let C be the event that the test *says* that the maximum level is exceeded.

What we want is then \(P(A^c|C^c)\)

So first we need the probability of C, P(C), which is the probability that the test says that the levels are exceeded. Do you know how to get that from our tree diagram?
(hint: total probability theorem)

Answer 12

no...

Answer 13

well if the levels are not exceeded (not A and B), and the test is accurate, we get C
also, if the levels are exceeded (A and not B), and the test is inaccurate, we get C
this means that C occurs when not A and B, or A and not B, which is this on our diagram:

Answer 14

er, I meant *not* C above (that the test says the levels are not exceeded) instead of C

Answer 15

the bottom of the tree arent those 2 the exact same thing but different wording

Answer 16

you mean not A and B and not A and not B are the same?

Answer 17

Yes

Answer 18

no, if not A and B, that means that the levels are not exceeded, and that the test is accurate, which means that the test correctly indicates that the limit is not exceeded (event not C). if not A and not B, then the limit is not exceeded, but the test is inaccurate, and so mistakenly thinks that it is (event C)

Answer 19

Oh, ok my bad.

Answer 20

no worries
so do you see how to calculate P(not C) yet?

Answer 21

I am probably such a hassle for you. I am sorry... No I do not.

Answer 22

not at all, I am a bit rusty as well
let's look at the various ways that even not C can occur. Here is one:what is the probability of this event?

Answer 23

event*

Answer 24

The black shaded one?

Answer 25

.428?

Answer 26

yes, that is one way
if A (that the limit is exceeded) and not B (the test is inaccurate) then not C (the test says the limit is not exceeded)

your answer is not correct, how did you get it?

Answer 27

.15/.35

Answer 28

or multiply them and get .0525?

Answer 29

yes, the first branch is P(A), and the second branch is P(B|A), so we want
P(B)=P(A)*P(B|A)

Answer 30

How would we get that

Answer 31

multiplying them, like you said

Let me try to simplify the situation a bit because I think I have confused you.

Let A=the levels are exceeded
Let B=the test says that levels are exceeded

Answer 32

I did multiply .15*.35

Answer 33

yes I know you did eventually, but I hope I am not confusing you
I was a little shaky on setting up this problem at first

Answer 34

No I am kind of getting it now.

Answer 35

ok, so the other way the test can say that the levels are not exceeded (event C) is if the levels are not exceeded, and the test is correct, which is this eventwhat is the probability of this event?