# What is the probability that the test shows that a person is lying?

When a person is lying the test says that the person is lying 90% of the time. When a person is telling the truth the test confirms this 88% of the time. Assume that this test is administered to everyone who applies for a certain job. Most people have no reason to lie and only 1% of applicants actually lie during the polygraph.

• What did you do? Did you try using Bayes theorem? – AspiringMat Dec 13 '16 at 6:03
• @AspiringMat I have tried Bayes theorem but that doesn't seem to work – B. smith Dec 13 '16 at 6:13
• I honestly do not know where to start now – B. smith Dec 13 '16 at 6:13

The test can show that the person is lying when he tells either the truth or a lie. We thus use the conditional probability technique. Let $L$ be the event the person is lying, we have $P(L) = 0.01$. Let $A$ be the event where the test says the person is lying.

When a person is lying the test says the truth with a probability of $0.90$, thus $P(A\mid L)$ is given. A person also tells the truth with probability $P(\bar L) = 1-P(L) = 0.99$. But when a person is telling the truth the test says the truth only with a probability of $0.88$. In $12$% of the cases it says the person is lying. Thus, we have the total probability as $P(A) =$ P(person is lying)P(Test says he is lying) + P(person says the truth)P(Test says he is lying) = $0.01(0.90)+0.99(0.12) = 0.1278$. Hope it helps.

• ok thanks that helps out a lot. For this question "Given that the test says a person is lying, what is the probability that the person is, in fact, telling the truth", would you use Bayes Theorem? – B. smith Dec 13 '16 at 7:17
• hmm. how would you solve it then? – B. smith Dec 13 '16 at 7:30
• Sorry, we have to use Bayes only. I didn't read the question properly. – Rohan Dec 13 '16 at 7:31

It's not Bayes, it's just straight conditioning. Let $A$ be the event "the person is lying", $B$ the event "the test says the person is lying". You were given $\mathbb P(A)$, $\mathbb P(B|A)$ and $\mathbb P(B^c|A^c)$ (where ${}^c$ is "complement").
Use the formula $$\mathbb P(B) = \mathbb P(B|A) \mathbb P(A) + \mathbb P(B|A^c) \mathbb P(A^c)$$

Another approach. Suppose $10000$ people apply. How many people are telling the truth? How many of them test as truth-telling? How many of them test as lying?

How many people are lying? How many of them test as truth-telling, and how many test as lying?

So out of the original $10000$ applicants, how many tested as lying? (Just for kicks, how many of those who tested as lying were actually lying?)