(I got a bit carried away writing this answer, sorry if I've gone into too much detail.)
The point of word problems is to see if you understand what the numbers mean. The difficulty is trying to get the mathematical data out of the wordy description. The numbers won't necessarily be given to you in the same order that you need to used them, and they might throw in some irrelevant details just to confuse you.
(Also, you shouldn't use Bayes for any of this.)
The best place to start is at the "root" of the tree with (A) and (B). At the moment I think you have (A) and (B) wrong. The point (A) is where we need to write the probability that the applicant tells the truth. The point (B) is where we need to write the probability that the applicant is lying. Either they're telling the truth or they're lying, so we know that (A)+(B)=100%.
The idea with a probability tree is that we deal with things starting at the root and working toward the branches (i.e. working from left to right). So (A) and (B) are the probabilities that the applicant is telling the truth or lying before we take the information from the lie detector into account. The stuff with the lie detector is dealt with on the next level of branches (i.e. C, D, E and F). So the bit of data you need for (A) and (B) is what they've told you at the end of the question:
the company has evidence to suggest that 9% of the applicants are lying.
This means that the company knows before using the lie detector that 9% of their employees are going to be lying. So (B) (the probability that they're lying) is 9%, and so (A) is 100%-9% which is 91%.
Next we deal with (C) and (D). These are on the branches coming out of (A), so they're the probabilities for the lie detector to say "Truth" or "Lie" given that the applicant is telling the truth (because the (A) branch is the one for when the person tells the truth). These come from the second number they've told you in the question
[The lie detector] incorrectly identifies 25% of true statements as lies.
So the probability it says "lie" given that the person is telling the truth, is 25%. So (C) is 25% and so (D) is 75%.
Finally we need (E) and (F). These are in the (B) branch of the tree, so we need the probabilities for the lie detector to say "Truth" or "Lie" given that the applicant is lying (because the (B) branch is the one for when the person lies). I'll let you work these ones out.
Notice that we didn't need Bayes' Theorem (or any formulas at all). This is because we were working up the tree from the root to the branches. Bayes' Theorem is for when you have to work down the tree in the opposite direction. In this problem it will tell you things like the probability that
they're telling the truth given that the machine says "Truth"
This is backward of what we worked out for (D) which is the probability that
the machine says "Truth" given that they're telling the truth.
You use the Bayes' formula for when you have to work out something like this that's going in the wrong direction. You'll probably have to do this later in the question.
Best Answer
"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"?