This problem continues the Focus Problem. The solution involves applying several basic probability rules and a little algebra to solve an equation.

(a) If the polygraph of Problem 27 indicated that 30% of the questions were answered with lies, what would you estimate for the actual percentage of lies in the answers? Let B = event detector indicates a lie. We are given P (B) = 0.30. Let A = event person is lying, so A^c = event person is not lying. Then

P (B) = P (A and B) + P (A^c and B)

P (B) = P (A) P (B | A) + P (A^c) P (B | A^c)

Replacing P (A^c) by 1 – ­ P (A) gives

P (B) = P (A) × P (B | A) + [1 – P (A)] × P (B | A^c)

Substitute known values for P (B), P (B | A), and P (B | A^c) into the last equation and solve for P (A).

(b) If the polygraph indicated that 70% of the questions were answered with lies, what would you estimate for the actual percentage of lies?