# Thread: Probability problem-urgent, need it for monday morning!

1. ## Probability problem-urgent, need it for monday morning!

new analytical method to detect pollutants in water is
being tested. This new method of chemical analysis is
important because, if adopted, it could be used to detect
three different contaminants: organic pollutants volatile
solvents and chlorinated compounds, instead of having to
use a different test for each pollutant.
The makers of the test claim that it can detect high levels of
organic pollutants with 99.35% accuracy, volatile solvents
with 99.15% accuracy and chlorinated compounds with
82.7% accuracy. If a pollutant is not present, the test does
not signal. Samples are prepared for the calibration of the
test and 50% of them are contaminated with organic
pollutants, 32% with volatile solvents and 18% with traces
of chlorinated compounds.
A test sample is randomly chosen.
(i) What is the probability that the test will signal?
(ii) If a test signals, what is the probability that
chlorinated compounds are present.

2. Originally Posted by Andreamet
new analytical method to detect pollutants in water is
being tested. This new method of chemical analysis is
important because, if adopted, it could be used to detect
three different contaminants: organic pollutants volatile
solvents and chlorinated compounds, instead of having to
use a different test for each pollutant.
The makers of the test claim that it can detect high levels of
organic pollutants with 99.35% accuracy, volatile solvents
with 99.15% accuracy and chlorinated compounds with
82.7% accuracy. If a pollutant is not present, the test does
not signal. Samples are prepared for the calibration of the
test and 50% of them are contaminated with organic
pollutants, 32% with volatile solvents and 18% with traces
of chlorinated compounds.
A test sample is randomly chosen.
(i) What is the probability that the test will signal?
(ii) If a test signals, what is the probability that chlorinated compounds are present.
Strewth, what a mouthfull. The first thing you absolutely have to do is break down all this gunk into clear mathematical statements:

Test gunk:

Pr(O.P. signal | O.P. present) = 0.9935

Pr(V.S. signal | V.S. present) = 0.9915

Pr(C.C. signal | C.C. present) = 0.827

Pr(no signal | no pollutant present) = 1

Calibration gunk:

Pr(O.P. present) = 0.5

Pr(V.S. present) = 0.32

Pr(C.C.) = 0.18

Now you absolutely have to give a clear mathematical statement of what you're trying to calculate:

(i) Pr(signal) = ?

(ii) Pr(C.C. present | signal) = ?

Then you use what you know to solve for ?

(i) Pr(signal) = Pr(O.P. signal | O.P. present) Pr(O.P. present)
+ Pr(V.S. signal | V.S. present) Pr(V.S. present) + Pr(C.C. signal | C.C. present) Pr(C.C. present) = ......

I get 0.96289, that is, a 96.289% chance of a signal.

-----------------------------------------------------------------------------------------------------------

(ii) Pr(C.C. present | signal) = $\frac{\text{Pr(C.C. present and C.C. signal)}}{\text{Pr(signal)}}$

Note that Pr(C.C. present and signal) = Pr(signal and C.C present) = Pr(signal | C.C. present) Pr(C.C. present) = .....

I get 0.154597, to the stated accuracy, that is, a 15.4597% chance.

But you should double check my arithmetic .....