We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
If you add up the probabilities for all the outcomes given X, i.e., P(C|X) and P(~C|X), then they have to add to one. Thus start by omitting the denominator, compute all the outcomes, add them up, and divide everything by the sum. The article actually does that.
So there's a total chance of 89.6% of a positive result, whether or not you really had cancer, and 10.4% of a negative result? I don't think I understand this at all yet. Does the answer mean that, if you get a positive result, you still have only a 7.6% probability of really having cancer? That doesn't sound right.