To solve this problem, let's break it down step by step:
1. Identify the given probabilities:
- The probability that the land has oil: [tex]\( P(\text{Oil}) = 0.45 \)[/tex]
- The accuracy of the test kit is [tex]\( 80\% \)[/tex], which means:
- It correctly identifies oil when there is oil: [tex]\( P(\text{Test positive} \mid \text{Oil}) = 0.80 \)[/tex]
- It correctly identifies no oil when there is no oil: [tex]\( P(\text{Test negative} \mid \text{No oil}) = 0.80 \)[/tex]
2. Calculate the probability that the land has no oil:
[tex]\[
P(\text{No oil}) = 1 - P(\text{Oil}) = 1 - 0.45 = 0.55
\][/tex]
3. Determine the probability that the test shows no oil given that there is no oil:
- The accuracy for correctly identifying no oil is [tex]\( P(\text{Test negative} \mid \text{No oil}) = 0.80 \)[/tex]
4. Calculate the joint probability that the land has no oil and the test shows no oil:
[tex]\[
P(\text{No oil and Test negative}) = P(\text{No oil}) \times P(\text{Test negative} \mid \text{No oil})
\][/tex]
[tex]\[
P(\text{No oil and Test negative}) = 0.55 \times 0.80
\][/tex]
[tex]\[
P(\text{No oil and Test negative}) = 0.44
\][/tex]
Thus, the probability that the land has no oil and the test shows no oil is [tex]\( 0.44 \)[/tex].
The correct answer is:
C. 0.44
