Question

1. Assume that you are interested in the likelihood that a stream violates water quality regulations. Assume 15% of streams are impaired. Also assume that that there is a 90% chance of detecting an impaired stream if it really is impaired (10% chance it will not be detected). Also assume that 5% of streams that are healthy are erroneously tested as impaired.

Fill-in the decision table with following conditional probabilities.

	Dectected	Missed
Healthy
Impaired

a. What is the probability that a stream will be detected as impaired?

b. Calculate the probability that a stream is impaired if a test comes back positive? Show your work

c. Calculate the probability that a stream is actually healthy if a test comes back negative? Show your work

Answer 1

Let H and I be the events that the stream is healthy and impaired respectively.

Also, let D and N be the events that impairment is detected and not detected respectively.

Since given that 15% stream are impaired.

P(I) = 0.15

Probability that a stream is healthy =P(H) = 1-P(I) = 1-0.15 = 0.85

Probability of defecting an impaired stream if it is impaired = P(D|I) = 0.90

Probability of not detecting an impaired stream given it is impaired = P(N|I) = 0.10

Probability that it is healthy and detected as impaired = P(HI) = 0.05

So,

So,

So the decision table will be :

	Detected	missed (not detected)	total
Healthy	0.05	0.80	0.85
Impaired	0.135	0.015	0.15
Total	0.185	0.815	1

Thus,

Probability that the stream will be detected as impaired = P(D) = 0.185