Final answer:
To find the probability of the soil being clay given a positive test result, we can use Bayes' rule. Given the probabilities of the different types of soil and the accuracy of the test in each soil type, we can calculate the probability using the law of total probability and Bayes' rule.
Step-by-step explanation:
To find the probability of the soil being clay given a positive test result, we can use Bayes' rule. Bayes' rule states that P(A|B) = (P(B|A) * P(A)) / P(B), where P(A|B) is the probability of event A happening given that event B has occurred, P(B|A) is the probability of event B happening given that event A has occurred, P(A) is the probability of event A happening, and P(B) is the probability of event B happening. In this case, event A is that the soil is clay and event B is that the test result is positive.
Given that the soil is clay, the test gives a positive result with 48% accuracy. Therefore, P(B|A) = 0.48. The probability of the soil being clay is P(A) = 0.21. To find P(B), we need to consider the probabilities of the test result being positive in each type of soil.
If the soil is rock, the test gives a positive result with 35% accuracy, so the probability of the test result being positive in rock soil is P(B|rock) = 0.35. Similarly, if the soil is sand, the test gives a positive result with 75% accuracy, so the probability of the test result being positive in sand soil is P(B|sand) = 0.75. We can calculate P(B) using the law of total probability: P(B) = P(B|rock) * P(rock) + P(B|clay) * P(clay) + P(B|sand) * P(sand).
Plugging in the given values, we have P(B) = 0.35 * 0.53 + 0.48 * 0.21 + 0.75 * 0.26. Now we can substitute the values into Bayes' rule:
P(clay | positive) = (P(positive | clay) * P(clay)) / P(positive) = (0.48 * 0.21) / P(B).
So the probability that the soil is clay given a positive test result is (0.48 * 0.21) / P(B).