Question

A false-positive in a polygraph test, that is, an individual who fails a lying-detector test even though they are telling the truth, has a probability of 15 %. Consider the experiment of administering a lie-detector or polygraph test (these are synonyms or words that have the same meaning in common usage) to a group of randomly selected people and recording false-positives. i) What is the probability that the first false-positive will occur when the third person is tested?

Answer 1

The probability of the first false-positive in a polygraph test occurring on the third person tested is approximately 10.8375%, calculated using the formula (1 - 0.15)^2 x 0.15.

Step-by-step explanation:

The question deals with the concept of geometric probability distribution, which is useful when an experiment is repeated independently until the first success occurs. In this case, a false-positive in a polygraph test (lie-detector test) is considered a 'success' although it's actually a misclassification, and we want to know the probability of this first success happening on the third trial.

Given that the probability of a false-positive is 15%, or 0.15, we can model the scenario as follows: the first two people tested must not have a false-positive (they are 'failures'), and then the third person tested must have a false-positive ('success').

The probability can, therefore, be calculated as:

P('Success' on 3rd test) = P('Failure' on 1st test) × P('Failure' on 2nd test) × P('Success' on 3rd test)
= (1 - 0.15) × (1 - 0.15) × 0.15
= 0.85 × 0.85 × 0.15
= 0.108375, or 10.8375%.

The probability that the first false-positive will occur when the third person is tested is approximately 10.8375%.