Given the ensuing well drilling data, answer the following.
• .30 (30%) probability of a productive well absent test information
• Pre-drilling ballistics test ‘s affirmative result correctly predicts a productive well .70 of the time
• Pre-drilling ballistics test’s negative result correctly predicts a non-productive well .80 of the time
9.1 Probability of a productive well given an affirmative ballistics test
9.2 Probability of a non-productive well given a negative ballistics test
No data is presented
oops, thank you drwls
Least to greatest -3, 0, .30
To find the probability of a productive well given an affirmative ballistics test (Question 9.1), we can use Bayes' Theorem.
Bayes' Theorem states that P(A|B) = (P(B|A) * P(A)) / P(B), where:
P(A|B) is the probability of event A occurring given that event B has occurred.
P(B|A) is the probability of event B occurring given that event A has occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.
In this case, event A is a productive well, and event B is an affirmative ballistics test.
We are given the following probabilities:
P(A) = 0.30 (30% probability of a productive well absent test information)
P(B|A) = 0.70 (affirmative ballistics test correctly predicts a productive well 70% of the time)
To find P(B), we need to consider both scenarios:
1. The ballistics test predicts a productive well correctly (true positive) with a probability of 0.70.
2. The ballistics test predicts a non-productive well incorrectly (false positive) with a probability of 0.30 (1 - 0.70).
So, P(B) = (P(A) * P(B|A)) + (P(not A) * P(B|not A))
P(B) = (0.30 * 0.70) + (0.70 * 0.30) = 0.21 + 0.21 = 0.42
Now we can calculate P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.70 * 0.30) / 0.42
P(A|B) = 0.21 / 0.42
P(A|B) = 0.50 (or 50%)
Therefore, the probability of a productive well given an affirmative ballistics test is 0.50 (or 50%).
To find the probability of a non-productive well given a negative ballistics test (Question 9.2), we can use a similar approach.
Let's define event C as a non-productive well and event D as a negative ballistics test.
We are given the following probabilities:
P(C) = 0.70 (probability of a non-productive well absent test information)
P(D|C) = 0.80 (negative ballistics test correctly predicts a non-productive well 80% of the time)
To find P(D), we need to consider both scenarios:
1. The ballistics test predicts a non-productive well correctly (true negative) with a probability of 0.80.
2. The ballistics test predicts a productive well incorrectly (false negative) with a probability of 0.30 (1 - 0.80).
So, P(D) = (P(C) * P(D|C)) + (P(not C) * P(D|not C))
P(D) = (0.70 * 0.80) + (0.30 * 0.20) = 0.56 + 0.06 = 0.62
Now we can calculate P(C|D):
P(C|D) = (P(D|C) * P(C)) / P(D)
P(C|D) = (0.80 * 0.70) / 0.62
P(C|D) = 0.56 / 0.62
P(C|D) ≈ 0.903 (or 90.3%)
Therefore, the probability of a non-productive well given a negative ballistics test is approximately 0.903 (or 90.3%).