Looking for counter-intuitive example for independence of random variables

Question

I am looking for a simple example of two independent discrete random variables that one would not expect to be independent because one knows that these two quantities have a causal relationship in real life.

Answer 1

Gambler's Fallacy is a pretty good example. People often expect past events to interfere with future independent events. For instance, if you flipped a coin ten times and got heads all ten times, you would expect to get tails the next time, even though the last ten flips have no effect on the next flip.

Answer 2

Along the lines of Michael's suggestion in the comments, we have the finger game of "odds and evens". Assuming each person's play is i.i.d.* at $50$-$50$ (by no means a guarantee, to be sure), the result is statistically independent of either person's play in isolation, but of course is completely determined by their joint play.

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*i.i.d. = identically and independently distributed (thanks to Bakuriu for clarifying this for the OP in the comments)

Answer 3

A classic basic example of events that seem dependent, but aren't, is when flipping two coins:

• $A$ is the event that coin 1 is heads.

• $B$ is the event that coin 1 and coin 2 are the same.

If you don't think about it too hard, it seems like the value of coin 1 is important or relevant when determining if coins 1 and 2 are equal, and therefore it is intuitively tempting to say $A$ is relevant to $B$ or in other words they are not independent.