Lie Detector!

Probability Level 2

James is going through a lie detector test. The probability that James will lie is $\frac{1}{4}$ . The lie detector will say he is lying when he is actually lying $\frac{4}{5}$ of the time. Otherwise, it will show he is telling the truth. The lie detector will also say he is lying when he is telling the truth $\frac{2}{5}$ of the time. Otherwise, it will show that he is telling the truth. Assuming that the lie detector only asks James one question, if the lie detector says that James is telling the truth, what is the probability that James is actually lying?

The answer is 0.1.

1 solution

Marta Reece
Jul 28, 2017

There are two ways the lie detector can come up with the claim that the answer is truthful.

James is lying, probability $\frac14$ , but this is reported as being true by the lie detector, probability $\frac15$ , to the combine probability of this happening being $\frac14\times\frac15=\frac1{20}$
James is telling the truth, probability $\frac34$ , and it is detected as such, probability $\frac35$ , to a combined probability of this event being $\frac34\times\frac35=\frac9{20}$

So these events happen in ratio of $1:9$ , or $\frac1{10}=\boxed{0.01}$ of the time the lie detector claims the statement to be correct it is actually a lie.

Lie Detector!

The answer is 0.1.

1 solution

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