Calculating probabilities of events

Question

Was going through past previous exam questions and came across this one:

A manufacturer of lie detectors is testing its newest design. It asks 300 people to lie deliberately and another 500 people to tell the truth. Of those who lied, the lie detector caught 200. Of those who told the truth, the lie detector accused 200 of lying. Let L describe the event that a person is a liar and N the event that the lie detector accuses a person of lying.

How do you calculate the following probabilities?

(1) P(L|N)

(2) P(L′|N′)

(3) Are the events L and N mutually exclusive?

(4) Are the events L and N independent?

Answer 1

Lets see the table below:

               Liar    Truth_Teller

 Detected      200     200

 Not_Detected  100     300

Hence:

$P(L)=300/(300+500)=3/8$

$P(N)=400/800=1/2$

$P(N,L)=200/800=1/4$

Now, from Bayesian rule:

(1) $P(L|N)=P(N,L)/P(N)=0.25/0.5=0.5$

(2) $P(L'|N')=P(L',N')/P(N')=300/800/(400/800)=0.75$

(3) No

(4) No