Poorly worded, but Simple Probability Question I'm having a discussion with my professor about the following probability problem:

Suppose that the probability that a driver is a male, and has at least one motor
  vehicles accident during a one-year period, is 0.12. Suppose that the corresponding
  probability for a female is 0.06. What is the probability of a randomly selected
  driver having at least one accident during the next 12 months?

My professor is insisting that the answer is 0.09, as the book "implies" that the sample space of the population is half male, and half female, and you would just take the average of the two probabilities.
I however, am attempting to argue that the probability is instead 0.18.  Following the probability axiom: P(A) = P(AB) + P(ABC), I propose that if you define A to to be the probability of being in a motor vehicle accident, and B to be the probability that the subject is a male, that the above formula should be applied to the problem to get the answer of 0.18.
I believe the difference comes down to wording.  If the problem had said "GIVEN that the driver was a male, the probability of having at least one motor vehicle accident is 0.12", I would agree with my professor's assessment.  But I would think that the wording of the problem clearly does not state that, but rather stating that P(AB)=0.12, and P(ABC)=0.06, giving my answer.
Can somebody help shed some light onto this?  Sorry if it seemed like I am rambling.
 A: To elaborate on both my comment and in general, I realize that in your interpretation, the probabilities I gave in my comment are impossible. The reason for this is that in your interpretation, you start out by picking a random driver.
This driver is male and has had a recent accident with probability 0.12, or the driver is female and has had a recent accident with probability 0.06.
Of course both of these probabilities cannot be larger that 0.5, because they are non-overlapping events.
Also, I agree that this is what the problem states, if you take it literally, and in this case, your answer is correct. If the problem was intended differently, then it should have been made more precise.
As a fun follow-up question, given this interpretation, you can ask: What is the probability of him having had a recent accident, if you choose a male driver? I get
$P(A|B) = P(AB)/P(B) = 0.12/0.5 = 0.24,$
which is of course a huge and not very realistic probability for this particular problem.
For the other interpretation, that for the set of male drivers, 12 % of them have had recent accidents, and likewise 6 % for female drivers, the problem of an arbitrary driver having had an accident is
$P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)=0.12\cdot 0.5 + 0.06\cdot 0.5 = 0.09.$
I'm sure that your professor chooses this interpretation of the problem, because he believes (through experience) that this is how the problem was intended, despite the poor wording.
