An actuary studying the insurance preferences of automobile owners makes the following conclusions:

What is the probability that an automobile owner purchases neither collision nor disability coverage?

- An automobile owner is twice as likely to purchase collision coverage as disability coverage.
- The probability that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage.
- The probability that an automobile owner purchases both collision and disability coverages is 0.15.

Ok, so I have:

The above answer is correct but my question is why can't I use De Morgen's law to change to and then just solve ? Is it because the two events aren't mutually exclusive or something? Thanks