Question

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 no oil and the test shows that it has oil?

Answer choices:

0.09

0.11

0.36

0.44

Answer 1

I know it's not A or C. But I tried the formula and got 1.778 but I don't know what to do from there.

Answer 2

Let D be the event that oil is detected
Let O be the probability that oil exists
We are given that \(P(O)=0.45\) and \(P(D|O)=0.8\)
We also know that
$$
P(D|O)=\frac{P(D\cap O)}{P(O)}
$$
Where \(\bar{O}\) is the event that there is no oil
Then
$$
P(D\cap O)=P(D|O)P(O)=0.80\times0.45
$$
We also know that
$$
P(D|O)+P(D|\bar{O})=1\\
\text{Which is the same as}\\
\frac{P(D\cap O)}{P(O)}+\frac{P(D\cap \bar{O})}{P(\bar{O})}=1\\
$$
We need \(P(D\cap \bar{O})\), the probability that oil is detected but none exists:
$$
\frac{P(D\cap O)}{P(O)}+\frac{P(D\cap \bar{O})}{P(\bar{O})}=1\\
\implies P(D\cap \bar{O})=\left(1-\frac{P(D\cap O)}{P(O)}\right )P(\bar{O})\\
=(1-0.8)0.55
$$

Does this make sense?