Question:
Yearly automobile inspections are required for residents of the state of Pennsylvania. Suppose that 18% of all inspected cars in Pennsylvania have problems that need to be corrected. Unfortunately, Pennsylvania state inspections fail to detect these problems 12% of the time. On the other hand, assume that an inspection never detects a problem when there is no problem. Consider a car that is inspected and is found to be free of problems. What is the probability that there is indeed something wrong that the inspection has failed to uncover?

My attempt:
Formulation
P = Problem
Ṗ = Complement of problem, i.e. no problem
Ḏ = Complement of detect, i.e. not detect
P(P) = 0.18
P(Ḏ | P) = 0.12

Solving
P(P | Ḏ) = ? is what we're trying to find

P(P ∩ Ḏ) = 0.12 0.18 = 0.0216

P(P | Ḏ) = P(P ∩ Ḏ) / P(Ḏ)

now lets try and find P(Ḏ)

P(Ḏ) = P(Ḏ ∩ P) + P(Ḏ ∩ Ṗ)
P(Ḏ) = P(Ḏ ∩ P) + P(Ṗ ∩ Ḏ)
P(Ḏ) = P(Ḏ ∩ P) + P(Ṗ) P(Ḏ | Ṗ)

we know that P(Ḏ | Ṗ) = 1 since from the question
"assume that an inspection never detects a problem when there is no problem"
means P(D | Ṗ) = 0
i.e. if given no problem, the probability of NOT detecting is 1

which leaves us with

P(Ḏ) = 0.0216 (found before) + 0.82 (= 1 - 0.12) 1
P(Ḏ) = 0.8416

so finally:

P(P | Ḏ) = P(P ∩ Ḏ) / P(Ḏ)
P(P | Ḏ) = 0.0216 / 0.8416
P(P | Ḏ) = 0.025665399

have I completely misunderstood the problem or am I bang on?

Thanks!