Coincidences are Improbable

I vote we abandon correlation does not imply causation in favor of connection does not imply direction. Or even better:

connection alone, direction unknown

But I'd like it best if we had a positive version.

Correlation does imply some sort of causal link.

For guessing its direction, simple models help you think.

Controlled experiments, if they are well beyond the brink

Of .05 significance will make your unknowns shrink.

Replications prove there's something new under the sun.

Did one cause the other? Did the other cause the one?

Are they both controlled by something already begun?

Or was it their coincidence that caused it to be done?

Occasionally I hear a song on the radio that I don't remember hearing in a while but that I had been thinking about only a day or two before. This surprised me, so I started thinking of possible explanations. My first thought was that I think about a lot of things and I hear a lot of songs on the radio so maybe there was bound to be a connection somewhere via the birthday paradox and that this was just more salient than the many times I had heard an unrelated song. I started making a mental note every time it happened, and even after considering the above, it still felt like it was happening too often to be pure coincidence. I currently think this happens because...

The two "direct" causal links are the only ones we would really call "causal" regarding A and B.

But I am a big fan of "correlation implies causation." It might not be between A and B specifically, but it means we've been able to detect something happening.

Sometimes even non-effects, when theory is strong enough, can indicate causation (though then the usual course of action is to control one of the paths to get an effect that you can talk about and publish). For example, you are about to eat an allergen, which you know causes side effects for you with p=1. You take Benadryl beforehand and have no side effects. There is no "effect" there (post state = pre state), but you can feel pretty sure Benadryl had a suppressing action on the allergen's effects (and then you would follow-up with experiments where you ate the allergen without Benadryl or took the Benadryl without eating the allergen to see the positive and negative effects separately).

Some event $D$ caused by both has been conditioned upon; new introductions have improbable attribute combinations because your friend seeks those combinations out.

This reads quite a bit like some sort of reverse-Barnum Effect, where instead of people trying to assign causality to vague descriptions / coincidences, they try to view linked (because of them) things as being coincidences. 

Coincidences are improbable, but improbable thing happens. Given any two events A and B, each one with probability 0.01/day, we expect that in about one day in 10000 both will happen. Seems pretty low, but what happens if there are several events A, B, C, ...., Z, each one with probability 0.01?

If we consider 25 distinct improbable events, there are 25X24/2 = 300 pair of events, and each of those pair has a probability of 10^(-4) of happening in a day. Therefore, in about 3% of days (that is, one a month) you will observe a coincidence, which is not anymore that low. 

In some of your examples, you are considering things that must have a well-defined cause (someone must have damaged your couch; something must be causing your skin irritation), so if the only thing new is the other event it is reasonable to guess a causal link (but what if you have a new dog and a new cat? Or if you changed lotion and you are also allergic to your new dog?). But in real life it is usually very difficult to insulate only one possible cause.

The latitude of the Great Pyramid of Giza coincides with the value of the speed of light in m/s, and the pyramids are roughly aligned with theposition of the Orion's Belt constellation in 12500 BC, and both these things are very unlikely, but I do not think there is something to uncover.

Coincidences can be evidence for correlation and therefore evidence for causation, as long as one remembers that evidence - like more things than most people feel comfortable with - are quantitative, not qualitative. A single coincidence, of even multiple coincidences, can make a causation less improbable - but it can still be considered very improbable until we get much more evidence.