Mineral Test Borings A geologist believes that the
probability that the rare earth mineral dalhousium occurs in the
Greater Toronto region is 0.001. If dalhousium is present in that
region, the geologist’s test borings will have a positive result 90%
of the time. However, if dalhousium is not present, a negative
result will occur 80% of the time.
(a) If a test is positive on a site in the region, find the probability
that dalhousium is there?
(b) If a test is negative on such a site, find the probability that
dalhousium is there?
To solve this problem, we will use Bayes' theorem. Bayes' theorem allows us to update the probability of an event based on new evidence.
Let's define some variables:
- A: The event that dalhousium is present in the Greater Toronto region.
- B: The event that a test is positive.
- A': The event that dalhousium is not present in the region.
- B': The event that a test is negative.
Given:
- P(A) = 0.001 (probability of dalhousium being present)
- P(B|A) = 0.9 (probability of a positive test given dalhousium is present)
- P(B|A') = 0.2 (probability of a positive test given dalhousium is not present)
- P(A') = 1 - P(A) = 1 - 0.001 = 0.999 (probability of dalhousium not being present)
(a) To find P(A|B) (probability of dalhousium being present given a positive test), we can use Bayes' theorem:
P(A|B) = (P(A) * P(B|A)) / P(B)
= (0.001 * 0.9) / P(B)
To find P(B), we need to consider all possible cases where a positive test can occur:
1. Dalhousium is present and the test is positive (P(A) * P(B|A))
2. Dalhousium is not present and the test is positive (P(A') * P(B|A'))
P(B) = P(A) * P(B|A) + P(A') * P(B|A')
= (0.001 * 0.9) + (0.999 * 0.2)
Now we can substitute the values into P(A|B):
P(A|B) = (0.001 * 0.9) / [(0.001 * 0.9) + (0.999 * 0.2)]
(b) To find P(A|B') (probability of dalhousium being present given a negative test), we can use Bayes' theorem again:
P(A|B') = (P(A) * P(B'|A)) / P(B')
= (0.001 * (1 - P(B|A))) / P(B')
To find P(B'), we need to consider all possible cases where a negative test can occur:
1. Dalhousium is present and the test is negative (P(A) * (1 - P(B|A)))
2. Dalhousium is not present and the test is negative (P(A') * (1 - P(B|A')))
P(B') = P(A) * (1 - P(B|A)) + P(A') * (1 - P(B|A'))
= (0.001 * (1 - 0.9)) + (0.999 * (1 - 0.2))
Now we can substitute the values into P(A|B'):
P(A|B') = (0.001 * (1 - 0.9)) / [(0.001 * (1 - 0.9)) + (0.999 * (1 - 0.2))]
Calculating these probabilities will give us answers for (a) and (b).