A 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 single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.7% accuracy, volatile solvents with 99.95% accuracy, and chlorinated compounds with 89.7% accuracy. If a pollutant is not present, the test does not signal. Samples are prepared for the calibration of the test and 60% of them are contaminated with organic pollutants, 27% with volatile solvents, and 13% with traces of chlorinated compounds. A test sample is selected randomly.
What is the probability that the test will signal? Textbook answer: 0.9847
If the test signals, what is the probability that chlorinated compounds are present? Textbook answer: 0.1184
Thanks in advance!
Solution
Let A1, A2 and A3 represent the events that the water contains contaminants, organic pollutants, volatile solvents, and chlorinated compounds respectively. ……………….............................................................................……………. (1a)
Let B represent the events that the test signals. ………........................................………………………………….. (1b)
Back-up Theory
If A and B are two events such that probability of B is influenced by occurrence or otherwise of A, then
Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)……....................................................….(2)
P(B) = {P(B/A) x P(A)} + {P(B/AC) x P(AC)}………………………………….................................................... .(3)
If A is made up of k mutually and collectively exhaustive sub-events, A1, A2,..Ak,
P(B) = sum over i = 1 to k of {P(B/Ai) x P(Ai)} ……………………...................................................………………....(4)
P(A/B) = P(B/A) x {P(A)/P(B)}……………………………..…….....................................................……………..…….(5)
Now to work out the solution,
Vide (1a) and (1b), the given data, in probability language would be:
P(A1) = 0.6 …………………………………………………..............................................……………………………. (6a)
P(A2) = 0.27 ……………………………………………............................................…………………………………. (6b)
P(A3) = 0.13 ……………………………………………………………………………...........................................…. (6c)
P(B/A1) = 0.997 ………………………………………………………………………………. .....................................(7a)
P(B/A2) = 0.9995 ………………………………………………………………………………. ...................................(7b)
P(B/A3) = 0.897 ………………………………………………………………………………...................................... (7c)
Part (a)
Probability that the test will signal
= P(B)
= {P(B/A1) x P(A1)} + {P(B/A2) x P(A2)} +{P(B/A3) x P(A3)} [vide (4)]
= (0.997 x 0.6) + (0.9995 x 0.27) + (0.897 x 0.13) [vide (6) and (7)]
= 0.9847 Answer
Part (b)
If the test signals, the probability that chlorinated compounds are present
= P(A3/B)
= P(B/A3) x P(A3)/P(B) [vide (5)]
= (0.897 x 0.13)/0.9847 [vide (7c), (6c) and Answer Part (a)]
= 0.1184 Answer
DONE
A new analytical method to detect pollutants in water is being tested. This new method of...
A 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 contaminates - organic pollutants, volatile solvents, and chlorinated compounds - instead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.3% accuracy, volatile solvents with 99.97% accuracy, and chlorinated compounds with 89.8%...
A 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 contaminates organic pollutants, volatile solvents, and chlorinated compounds instead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.4% accuracy, volatile solvents with 99.90% accuracy and ch onnated compounds with 89.89 accuracy...