The probability that a person is lying if the lie detector shows a positive reading is A) 90% (This is the closest option to the answer).
We shall use Bayes' theorem to find the probability that a person is lying given a positive reading from the lie detector.
Let P(L) = probability that a person is lying,
Let P(T) = probability that a person is telling the truth
Given:
Probability of a positive reading given a lie = P(Pos\L) = 95% (or 0.95)
Probability of a positive reading given the truth = P(Pos\T) = 10% (or 0.10)
Bayes' theorem is expressed as:
P(L/Pos) = P(Pos\L) .P(L) / P(Pos\L).P(L) + P(Pos\T).P(T)
Supposing an equal chance of lying and telling the truth initially (P(L) + P(T) = 0.5,
We shall now plug in the values:
P(L\Pos) = (0.95). (0.5) / (0.95). (0.5) + (0.10).(0.5)
P(L\Pos) = 0.475 / 0.525
P(L\Pos) = 0.9048 * 100 = 90.48
The probability that a person is lying if the lie detector shows a positive reading is 90.48%