Question

On a true-false test, the probability that a student knows the answer to a question is 0.7. If she knows the answer, she checks the correct answer; otherwise, she answers the question by flipping a fair coin.
(a) What is the probability that she answers a question correctly?
(b) Given that she answers the question correctly, what is the probability that she knew the answer?

Answer 1

H1 = { we know the answer }

H2 = { we don't know the answer and will flip a fair coin }

A ={ choosing the correct answer }

a) By the formula of total probability , we have

$P(A) = \sum P(H_i)*P(A|H_i)$

So P(A) = 0.7 * 1 + 0.3 * 0.5 = 0.85

Answer 1): 0.85

b) By formula of Bayes:

$P(H_i|A) = P(A*H_i)/P(A)= P(A|H_i)*P(H_i)/P(A)$

So for second question we have to find $P(H_1|A)$ :

$P(H_1|A) = P(A|H_1)*P(H_1)/P(A) = 1* 0.7 / 0.85 = 0.823529...$

Answer 2) 0.823529...