The specific example I gave is more due to treating random variables as if they're independant. For example, you're as likely to be off either way on A, and you're as likely to be off either way on B, so for each of those, you in fact gave the correct probability, but you're more likely to be off the same way on both than the opposite ways, so you have to correct more when you use them together.

But yes. Bayes' theorem is always the answer.