Question

Liar, liar! Sometimes police use a lie detector (also known as a polygraph) to help determine whether a suspect is telling the truth. A lie detector test isn’t foolproof—sometimes it suggests that a person is lying when they’re actually telling the truth (a “false positive”). Other times, the test says that the suspect is being truthful when the person is actually lying (a “false negative”). For one brand of polygraph machine, the probability of a false positive is $0.08$.
(a) Interpret this probability as a long-run relative frequency.
(b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Short answer

Part (a) The truth is that people lie about $8\%$ of the time, according to the machine.

Part (b) The truth is that people lie about $8\%$ of the time, according to the machine.

Step-by-step solution

Part (a) Step 1. Given Information

A false positive has a $0.08$ percent chance of occurring.

Part (a) Step 2. Concept Used

We can't foresee the outcomes of a chance process, yet they have a regular distribution over a large number of repetitions. According to the law of large numbers, the fraction of times a specific event occurs in numerous repetitions approaches a single number. The likelihood of a chance outcome is its long-run relative frequency. A probability is a number between $0$ (never happens) and $1$ (happens frequently) (always occurs).

Part (a) Step 3. Calculation

The fraction of times an answer occurs is called a relative frequency, and it is calculated by dividing each frequency by the total number of students in the sample.

That is correct $$\begin{gathered} =0.08x100 \\ =8\% \end{gathered}$$

This probability is calculated as a long-run relative frequency by using a polygraph machine on a group of people who are all telling the truth. The machine will say that the people are lying around $8\%$ of the time.

Part (b) Step 1. Explanation

The fraction of times an answer occurs is called a relative frequency, and it is calculated by dividing each frequency by the total number of students in the sample. Claiming someone is lying while they are stating the truth appears to be a more dangerous error than a false positive or false negative.