Answer:
The probability that the inhabitant spoke thruthfully given than the other one says so is 1/4.
Step-by-step explanation:
Let A be the event 'the first inhabitant speaks the truth' and B the event the person you asked speaks the truth'
We have that
P(A) = P(B) = 1/3
The second person will say that the first person speaks the truth if:
- Both are lying
- Both are saying the truth
The probability that both inhabitants lies is 2/3 * 2/3 = 4/9
The probability that both inhabitants speaks the truth is 1/3² = 1/9
Therefore, in this problem, we want to know P(A ∩ B | (A∩ B) ∪ (A^c ∩ B^c) )
(note that, since the second persons says that A didnt lie, then in order for it to be true then the second person have to also say the truth).
We know that P(A ∩ B) = 1/9 and P((A∩ B) ∪ (A^c ∩ B^c)) = 4/9 + 1/9 = 5/9. Using the bayes formula we have
P(A ∩B | (A∩ B) ∪ (A^c ∩ B^c) ) = P((A∩ B) ∪ (A^c ∩ B^c) | A ∩ B) * P(A∩ B)/ P((A∩ B) ∪ (A^c ∩ B^c))
Note that P((A∩ B) ∪ (A^c ∩ B^c) | A∩B) = 1 because the condition is more restrictive than the probability we are asking for, therefore
P(A ∩B | (A∩ B) ∪ (A^c ∩ B^c) ) = P(A∩ B)/P((A∩ B) ∪ (A^c ∩ B^c)) = (1/9) / (4/9) = 1/4.
The probability that the inhabitant spoke thruthfully given than the other one says so is 1/4.
