A Probability question on Bayes theorem

Question

I am struggling to understand a problem from https://docplayer.net/6566428-Probability-exam-questions-with-solutions-by-henk-tijms-1.html

Problem :

On the island of liars each inhabitant lies with probability 2/3 . You overhear an inhabitant making a statement. Next you ask another inhabitant whether the inhabitant you overheard spoke truthfully. Find the probability that the inhabitant you overheard indeed spoke truthfully given that the other inhabitant says so

Here is my Answer -

A = 1st person is actually truthful

B : 2nd person says 1st person is truthful

So we need P[A|B]

Bays theorem : $P[A|B] = \frac{P[B|A]P[A]}{P[B|A]P[A] + P[B|A^c]P[A^c]}$

So $P[A] = 1/3, P[A^c] = 2/3, P[B|A] = 1/3, P[B|A^c] = 2/3$

This gives me answer as 0.20

But, actual answer comes as 0.25.

Can somebody please help me to understand where am I wrong?

Answer 1

"On the island of liars each inhabitant lies with probability 2/3 . You overhear an inhabitant making a statement. Next you ask another inhabitant whether the inhabitant you overheard spoke truthfully. Find the probability that the inhabitant you overheard indeed spoke truthfully given that the other inhabitant says so."

Here's how I would do this: Imagine this happens 900 times. Then 600 times the first person lied, 300 times told the truth. Of the 300 times the first person told the truth, 100 times the second person says he told the truth, 200 times says he lied. Of the 600 times the first person lied, 400 times the second person says he told truth 200 times the second person says he lied.

So the second person says that the first person told the truth a total of 100+ 400= 500 times. Of those 500 times, the first person actually told the truth 100 times.

The probability the first person told the truth, given that the second person said he did is $\frac{100}{500}= \frac{1}{5}= 0.2$.

What makes you say that "the actual answer comes back as 0.4"?

Answer 2

Your answer seems right, though maybe we are both misunderstanding something.

Not using Bayes’s thereom, Imagine repeating this experiment $81$ times, with outcomes matching the given probabilities.

The statement is true $27$ times, and it’s (always correctly) described as true $9$ of those times.

The statement is false $54$ times, and it’s (always incorrectly) described as true $36$ of those times.

So the statement is described as true $9+36=45$ times, and the description is correct $9$ of those $45$ times, which is 20%.