A polygraph machine correctly identifies a lie 81% of the time, and incorrectly identifies a true statement as a lie 8% of the time. If a person being examined with the machine lies 10% of the time, what is the probability that a statement identified by the polygraph as a lie is actually true?
To solve this problem, we can use Bayes' theorem. Bayes' theorem allows us to calculate the probability of an event occurring given prior knowledge or information.
Let's define the following events:
A - A statement is identified as a lie by the polygraph.
B - The statement is actually true.
We are given the following information:
P(A|B) = 0.08 (The probability that the polygraph incorrectly identifies a true statement as a lie)
P(A|~B) = 0.81 (The probability that the polygraph correctly identifies a lie)
P(B) = 0.10 (The probability that a statement is actually true)
We want to calculate:
P(B|A) (The probability that a statement identified as a lie is actually true)
Now, using Bayes' theorem, we have:
P(B|A) = (P(A|B) * P(B)) / P(A)
To calculate P(A), we can use the Law of Total Probability:
P(A) = P(A|B) * P(B) + P(A|~B) * P(~B)
P(A|~B) is the probability that the polygraph correctly identifies a lie, which is 1 minus the probability of falsely identifying the statement as a lie:
P(A|~B) = 1 - 0.08 = 0.92
P(~B) is the probability that the statement is not true, which is 1 minus P(B):
P(~B) = 1 - P(B) = 1 - 0.10 = 0.90
Now we can substitute these values into the equation for P(A):
P(A) = (P(A|B) * P(B)) + (P(A|~B) * P(~B))
= (0.08 * 0.10) + (0.92 * 0.90)
Simplifying, we find:
P(A) = 0.008 + 0.828 = 0.836
Now we can substitute the values for P(A), P(A|B), and P(B) into the equation for P(B|A):
P(B|A) = (P(A|B) * P(B)) / P(A)
= (0.08 * 0.10) / 0.836
Calculating this expression, we find:
P(B|A) ≈ 0.00957
Therefore, the probability that a statement identified by the polygraph as a lie is actually true is approximately 0.00957, or 0.957%.