Answer to Question #345997 in Statistics and Probability for arty

Question

Engineers in charge of maintaining out nuclear fleet continually check for corrosion inside the pipes that are part of cooling system. The inside condition of the popes cannot be observed directly but a non-destructive test can give an indication of possible corrosion. The test is not infallible. The test has probability of 0.7 of detecting corrosion when it is present but it also has probability of 0.2 of falsely indicating internal corrosion. Suppose the probability that any section of pipe has internal corrosion is 0.1.

Answer 1

Let C="internal corrosion exists" and let D = "internal corrosion detected".

$P(C)=0.1,\; P(D|C)=0.7,\;and\; P(D|C')=0.2$

So, e the probability that corrosion exists given that the test detects corrosion is:

$P(C|D)=\frac{P(D|C)P(C)}{P(D|C)P(C)+P(D|C')P(C')}$ by Bayes′s Rule

$=\frac{0.7*0.1}{0.7*0.1+0.2*0.9}$ via given probabilities

$=0.28$

That is, a positive test result changes the probability of corrosion from 0.1 to 0.28.