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:

$\displaystyle p(c)=2p(d)$

$\displaystyle p(c\cap d)=0.15$

$\displaystyle =>p(c)*p(d)=0.15$

$\displaystyle 2p(d)^{2}=0.15$

$\displaystyle =>p(d)=\sqrt{0.075}$

$\displaystyle p(c)=2p(d)=2\sqrt{0.075}$

$\displaystyle p(c'\cap d')=>p(c')*p(d')$

$\displaystyle =>(1-2\sqrt{0.075})*(1-\sqrt{0.075})\approx 0.3284$

The above answer is correct but my question is why can't I use De Morgen's law to change $\displaystyle p(c'\cap d')$ to $\displaystyle p(c\cup d)'$ and then just solve $\displaystyle 1-p(c \cup d)=>1-(\sqrt{0.075}+2\sqrt{0.075})$? Is it because the two events aren't mutually exclusive or something? Thanks