13 Define F as failed in the diagnostic test that P(F)=0.08 and A as failed in the real test. Given a student was failed in the diagnostic test, the probability he/she is also failed P(A|F)=0.90. The conditional given the student was not failed in the diagnostic test and not failed in the real test is 0.8. (Bayes’ questions)

a. What is the probability given the student has passed the real test but failed in the diagnostic test?
b. What is the probability given the student has failed the real test that he/she failed in the diagnostic test?

To solve these Bayes' theorem questions, we need to understand the given information and apply the formulas accordingly.

Given:
P(F) = 0.08 (probability of failing the diagnostic test)
P(A|F) = 0.90 (probability of failing the real test given the student failed the diagnostic test)
P(not F and not A) = 0.8 (probability of not failing the diagnostic and real tests)

a. What is the probability given the student has passed the real test but failed in the diagnostic test?

Let's calculate P(A' and F'), where A' represents passing the real test and F' represents failing the diagnostic test.

Using Bayes' theorem, we have:
P(A' and F') = P(F') * P(A'|F') [Using the formula P(A and B) = P(A) * P(B|A)]

We know that:
P(F') = 1 - P(F) [Since F' represents not failing the diagnostic test]
P(A'|F') = 1 - P(A|F') [Since A' represents passing the real test]

Substituting these values, we get:
P(A' and F') = (1 - P(F)) * (1 - P(A|F'))

b. What is the probability given the student has failed the real test that he/she failed in the diagnostic test?

Let's calculate P(F|A), where F represents failing the diagnostic test and A represents failing the real test.

Using Bayes' theorem, we have:
P(F|A) = P(A) * P(F|A) / P(A) [Using the formula P(A|B) = P(A) * P(B|A) / P(B)]

We know that:
P(F|A) = P(F) [Probability of failing the diagnostic test given failing the real test is the same as the probability of failing the diagnostic test]
P(A) = 1 - P(A') [Probability of failing the real test]

Substituting these values, we get:
P(F|A) = P(A) * P(F) / (1 - P(A'))