This topic comes up here periodically, and I've linked to these articles before, but it's probably worth doing so again for those interested. I am a physician who was also puzzled by these matters, and wound up inventing a simple diagram that I use as a mental "widget" to manipulate the ideas of the 2x2 table in my head. The full text articles can be found at: https://www.researchgate.net/profile/Kevin_Johnson40
P.S. I'd love to have someone animate this in Javascript!
I had to struggle with this concept as a physician trying to understand the medical literature. Eventually I came up with my own way of thinking dynamically about it. I called this the "two by two diagram". You can find the full text of my articles here:
https://www.researchgate.net/profile/Kevin_Johnson40
As a matter of fact, I would very much like to get this programmed into a Javascript version I can make freely available to people, but am not clear on how to go about getting that done. I have a version in MATLAB but that is not practical for most people.
There is a limited demo there as well.