A lab test to detect a particular contaminant in drinking water has probability

Question

A lab test to detect a particular contaminant in drinking water has probability 0.75 of coming up
positive when the contaminant is present, and probability 0.02 of coming up positive when the
contaminant is not present. This particular contaminant is fairly rare and is generally found in only 5%
of water specimens.

(a) What is the probability that a randomly selected water specimen would test positive? Show your
work using proper probability notations.
(b) You receive a positive test result for a water specimen you submitted for a routing inspection.
Given this information, what is the probability that the specimen actually contains this particular
contaminant? Show your work using proper probability notations.

Explanation / Answer

Let the event S = contaminant is in the water sample. T = The water sample tested positive for contaminant P(S) = 0.05 (Since contaminant is present in 5% of water specimens) a) P(T) = P(S)*P(T|S) + P(S')*P(T|S') = 0.05*0.75 + (1-0.05)*0.02 = 0.0565 b) Given, P(S|T) = ? (given question is actually "event S given event T is true") P(S|T) = P(S n T)/P(T) = P(S)*P(T|S)/P(T) = 0.05*0.75/0.0565 = 0.6637

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