Note that this question is already asked. I do have solution. But I want to know specifically what's the flaw in following approach.
Question: You are lost in the National park of Kabrastan. The park population consists of tourists and Kabrastanis. Tourists comprise two-thirds of the population the park and give a correct answer to requests for directions with probability $\frac{3}{4}$. The air of Kabrastan has an amnesaic quality, however, and so the answers to repeated questions to tourists are independent, even if the question and the person are the same. If you ask a Kabrastani for directions, the answer is always wrong. Suppose you ask a randomly chosen passer-by whether the exit from the park is East or West. The answer is East. You then ask the same person again, and the reply is again East. What is the probability of East being correct?
Following is my approach which I know is wrong but I don't know what's the flaw in that.
$A: \text{East is indeed the right way to exit.}$
$B: \text{Passer answers east two times.}$
Let's condition upon weather passer chosen is tourist or Kabrastani.
Case 1: Passer chosen is tourist.
then, $P(A|B) = \frac{P(B|A)*P(A)}{P(B|A)*P(A) + P(B|A')*P(A')} = \frac{\frac{9}{16}*\frac{1}{2}}{\frac{9}{16}*\frac{1}{2} + \frac{1}{16}*\frac{1}{2}} = \frac{9}{10}$
Case 2: Passer chosen is Kabrastani.
here, $P(A|B) = 0$ because, we know that is Kabrastani gives answer as east it is definitely not the right way to go.
So, required probability = $P(\text{passer chosen is tourist})*P((A|B)|\text{passer chosen is tourist}) + P(\text{passer chosen is Kabrastani})*P((A|B)|\text{passer chosen is Kabrastani})$
= $\frac{2}{3}*\frac{9}{10} + \frac{1}{3} * 0 = \frac{3}{5}$
But answer given is $\frac{1}{2}.$