Jason inherited a piece of land from his great-uncle. Owners in the area claim that there is a 45% chance that the land has oil. Jason decides to test the land for oil. He buys a kit that claims to have an 80% accuracy rate of indicating oil in the soil. If the test predicts that there is no oil, what is the probability after the test that the land has oil?
0.09
Jason inherited a piece of land from his great-uncle. Owners in the area claim that there is a 45% chance that the land has oil. Jason decides to test the land for oil. He buys a kit that claims to have an 80% accuracy rate of indicating oil in the soil. What is the probability that the land has oil and the test predicts that there is no oil?
0
0.09
0.11
0.44
To determine the probability after the test that the land has oil, we can make use of Bayes' theorem.
Let's define the following events:
A: The land has oil
B: The test predicts no oil
We are given the following probabilities:
P(A) = 0.45 (the initial probability that the land has oil)
P(B | A) = 0.2 (the probability that the test predicts no oil given that the land actually has oil)
P(B | not A) = 0.8 (the probability that the test predicts no oil given that the land does not have oil)
We are looking for the probability P(A | B), the probability that the land has oil given that the test predicts no oil.
Using Bayes' theorem, we have:
P(A | B) = P(B | A) * P(A) / P(B)
To find P(B), we need to consider all possible scenarios where the test predicts no oil. This includes cases where the land has oil and the test correctly predicts no oil, as well as cases where the land does not have oil and the test incorrectly predicts no oil. We can write this as:
P(B) = P(B | A) * P(A) + P(B | not A) * P(not A)
To find P(not A), we can subtract P(A) from 1:
P(not A) = 1 - P(A)
Plugging in the given values into the equations, we have:
P(B) = 0.2 * 0.45 + 0.8 * (1 - 0.45)
= 0.09 + 0.8 * 0.55
= 0.09 + 0.44
= 0.53
Now we can substitute P(B) and the other values back into the equation for P(A | B):
P(A | B) = P(B | A) * P(A) / P(B)
= 0.2 * 0.45 / 0.53
≈ 0.16981
Therefore, after the test predicts no oil, the probability that the land has oil is approximately 0.16981 (or approximately 16.98%).
To find the probability of oil in the land after the test, we need to use Bayes' Theorem, which relates conditional probabilities.
Let's define the following events:
A: The land has oil.
B: The test predicts no oil.
We are given the following probabilities:
P(A) = 0.45 (the original claim that there is a 45% chance the land has oil)
P(B|A) = 0.2 (the accuracy rate of the test indicating no oil)
P(B|A') = 1 - P(B|A) = 0.8 (the accuracy rate of the test indicating oil)
Using Bayes' Theorem, we can calculate the probability of oil in the land after the test:
P(A|B) = (P(B|A) * P(A)) / P(B)
To calculate P(B), we can use the law of total probability. There are two possibilities: either the land has oil (A) or it doesn't (A').
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
= 0.2 * 0.45 + 0.8 * (1 - 0.45)
= 0.09 + 0.36
= 0.45
Substituting these values into Bayes' Theorem:
P(A|B) = (0.2 * 0.45) / 0.45
= 0.09 / 0.45
= 0.2
Therefore, the probability of the land having oil after the test predicts no oil is 0.2, or 20%.