[extropy-chat] Coin Flip Paradox
jef at jefallbright.net
Wed Jan 31 16:08:48 UTC 2007
[Top-posting since it would be confusing to attempt comments within the
Gordon, in real life I'd expect Jef to consistently refer you back to
the principle that equivalent states of information yield equivalent
properties (and likely mention the relevance of limited context in any
real-world scenario of decision-making.)
The scenarios contributed by Stuart are very much to the point and
should be helpful.
As to your scenario of the possibly bent coin, it should be helpful if
you would imagine how you would bet if your survival depended on
minimizing the error magnitude of the subsequent trials and you weren't
allowed to claim "no information, so no bet."
> Gordon: "Say there, Jef! Can you help explain something to me? I
> know much about probability theory and I was hoping you could help
> me out."
> Jef: "Sure! How can I help you?"
> Gordon: "Let's say I have a coin hidden in my pocket. I retrieved
> coin from my garbage disposal, so I should tell you it might be
> bent and
> unfair, or it might be normal and fair. Are you justified in
> believing or
> assuming it is fair?"
> Jef: "What a silly question, Gordon! Of course not! I would need to
> inspect your coin or see some other evidence before I would be
> in commenting in any way about the fairness or unfairness of that
> coin in
> your pocket. Until then I have no justification for believing or
> it is fair, and no justification for believing or assuming it is
> not fair."
> Gordon: "Okay, that's what I thought. Just wanted to be sure.
> Thanks for
> the lesson."
> Jef: "You're welcome!"
> (The ghost of Pierre-Simon Laplace enters the room and taps Jef on
> Laplace: "Excuse me, but I couldn't help overhearing that
> Jef: "Huh? Who are you? Borat?"
> Laplace: "No, I am the one-and-only Pierre-Simon Laplace,
> Magnificent High
> Priest of the Glorious Classical School of Probability! I am here
> enlighten you about the Beautiful Principle!"
> Jef: "What Beautiful Principle?"
> Laplace: "My devoted followers called it the Principle of
> Reason. You moderns call it the Principle of Indifference. The
> states that if you have no reason to expect one outcome over
> another, the
> outcomes are equiprobable."
> Jef: "Wow! That really IS beautiful! It's BEAUTIFUL and it's
> ELEGANT, so
> it must be a TRUE PRINCIPLE OF LOGIC!"
> Laplace: "That's what I thought, too. Good luck with it, Jef. I
> have to go
> (The ghost of Laplace leaves the room.)
> Jef: "Say there, Gordon! Come on back here! I have something new to
> Gordon: "Great, what is it?"
> Jef: "I was wrong before when I said I have no justification for
> that coin in your pocket is fair. According to the Beautiful
> Principle of
> Indifference, I am justified in assuming the coin IS fair because I
> no reason to believe it is NOT fair. I heard this from a very
> source: Pierre-Simon Laplace!"
> Gordon: "Laplace? Reliable? But wasn't he wrong about the
> Jef: "Maybe, but this isn't about that. It's about The Beautiful
> Principle, Gordon! Don't you see? It's BEAUTIFUL and it's ELEGANT
> and it's
> Gordon: "Let me get this straight: a few minutes ago you told me
> you were
> NOT justified in believing or assuming anything about the fairness
> of the
> coin in my pocket, and now you're telling me you ARE justified in
> the coin is fair, *despite having obtained no new knowledge about
> the coin
> itself*. Is that right?
> Jef: "Right. It's about the Principle, Gordon. It's BEAUTIFUL. You
> be glad I told you about it! You need only dwell on its magnificent
> for a few moments, and then you too will have THE POWER."
> Gordon: "If you say so..."
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