Let's say I have a factory producing 28% defective bottles. A defective bottle has a 66.5% chance of breaking when dropped on the ground and a non-defective bottle only has a 9.5% chance to break.
Given that I dropped a randomly picked bottle on the ground 4 times and it didn't break (its chance to break doesn't increase with every drop), what is the chance for that bottle to be defective?
This is a homework question and I attempted to use Bayes' rule such as
$ P(Defective | IntactX4) = {P(IntactX4|Defective)P(Defective)\over P(IntactX4)} $
$ P(Defective | IntactX4) = {(1-0.665)^4 * 0.28\over (0.28*(1-0.665)+(1-0.28)*(1-0.095))^4} $
Therefore
$ P(Defective | IntactX4) \approx0.011422987612 $
However, this does not appear to be the correct answer. I wonder if there is something wrong with the way I use Bayes' rule. Can anyone shed some light?