Solve This!
Peter Gleeson
92

Hi Peter, you nicely touched a topic that is very difficult to comprehend. And that’s probabilistic interpretation of independence. Allow me to show you slightly different explanation of the puzzle.

When one event clearly doesn’t influence the other, are they independent? Not really. When one event clearly influences the other, are they dependent? Not necessarily.

Regarding to your story, the two dice are not independent. In fact, red and blue die are independent under prior, just before you toss. However, later on, when you show me a die, they are not independent under posterior.

Let me be precise. Before you toss, I can reasonably expect P(red=6)=1/6, P(blue=6)=1/6 and P(red=6&blue=6)=1/36, which means red and blue die are independent in a chance experiment with 36 possible outcomes.

However, under posterior condition when I know a die (not red nor blue) has 6 points, the chance experiment has only 11 outcomes. Interestingly P(red=6)=6/11, P(blue=6)=6/11, P(red=6&blue=6)=1/11, therefore when you provide me an information about one die, I can infer knowledge about the other die. The dice are in fact dependent!

The intuition fails only because the underlying chance experiment has changed (sneaky!). When you make the independence claim at the beginning, be careful to denote the chance experiment you are referring to. Once you become deus ex machina in the game (choosing only right outcomes), your decision silently changes the rules and beliefs no longer hold.

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