Independence dependent on conditioning.
I used to tutor high-school students. One question I liked to ask: “Are two incompatible events independent?”. Bringing those two notions together helped to clarify both of them. In some cases, it could also lead to a profound state of confusion, which is always pleasant to observe.
Now, I’d like to remind you (and future me) of the perils of intuition in probabilities. For instance, conditioning can break independence. If you roll two dice, one fushia and one vert de gris, the events A: “Ace on fushia” and B: “Ace on vert de gris” are independent. Not so much under the condition C: “Sum is even”.
One could reply: “That’s not surprising. Conditioning updates our knowledge of the world and new connections can be established. That’s the basis of bayesian inference.” That would be a pedantic yet appreciable comment. The interesting twist is that dependent events can be made independent under a certain context. In this case we are somehow losing and loosing information.
Another possible oddity: A can be independent from both B and C, but dependent with B ⋂ C (or dependent with B ⋃ C for that matter).
Final note. Pretty much like correlation, dependency (or absence thereof), being a numerical property, can arise by numerical coïncidence. Everything is connected. Yet some connections are more meaningful than others. It’s still up to the humanoïds to find which ones.
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