I'm pretty sure pigeons don't know probability. Instead, they have a randomizing algorithm.
http://en.wikipedia.org/wiki/Randomized_algorithm
This allows them to discover that the grass is sometimes greener on the other side, in other words they discover a pattern in a random event, after all it's not guaranteed that there's always food in the other door, there's no direct and deterministic cause and effect, instead it's just more likely. Hence, for the pigeons, discovering a pattern in randomness is economically useful, not a folly as some might suggest. (I'm assuming the pigeons learned to increase their choice of the other door, but maybe they simply kept their randomizing ratio the same and weren't smart enough to learn).
What makes it more interesting is that the famous mathematician Paul Erdos, who thought he knew probability theory perfectly, was not at all convinced, until he looked at a Monte Carlo sim. A Monte Carlo sim is a brute force method, it simply spits out the possible results. The mathematician Paul Erdos instantly recognized a pattern in the random data, just as the pidgeons had recognized a pattern in the data they collected, namely that changing ones choice doubled ones odds, but Erdos was able to also correct the mistake he had made in his earlier judgement using probability theory. The moral of the story is that it is sometimes useful to run a Monte Carlo analysis, or simply start collecting data, and confirm that the patterns revealed are consistant with ones understanding of probability theory.
http://en.wikipedia.org/wiki/Monte_Carlo_method
Mickey's suggestion that one try dealing the cards to oneself first and watching the patterns is a form of the Monte Carlo method.
Other links that might be found useful:
http://en.wikipedia.org/wiki/Monte_Hall_Problem
http://en.wikipedia.org/wiki/Empirical_probability
http://en.wikipedia.org/wiki/Classical_probability
http://en.wikipedia.org/wiki/Determinism
http://en.wikipedia.org/wiki/Superdeterminism
--- In vpFREE@yahoogroups.com, David Silvus <djsilvus@...> wrote:
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> "So, in the Monty Hall problem, can we perceive a pattern in the remaining two doors? When 1 and 2, more likely to be 1, when 2 and 3 more likely to be 2, etc.? I don't think the pigeons did - they learned to make a choice based on probability, not patterns, and I didn't see evidence that they then chose the correct door with any edge."
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> What's the difference between "probability" and "pattern" though? The "pattern" is that your "probability" of reward is twice as great if you change your initial choice than not. If one group opts to change and ther other group opts to stand pat, the first group has recognized that pattern while the 2nd group has not.
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> Honestly, as addressed by someone else before, the term " pattern" really needs to be defined more tightly for this discussion. I readily agree that my usage of the term in the preceeding paragraph does not comport with its definition as used in mathematics. If that is the definition being used for the term, however, then the discussion becomes far less interesting on the macro and micro level (at least to me).
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> To: vpFREE@yahoogroups.com
> From: allen-walker@...
> Date: Wed, 5 Jan 2011 21:18:28 +0000
> Subject: [vpFREE] Re: Best Randomness Analogy Contest
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> Gee, Frank, did you think that the religion aspect would turn out to be the definition of randomness?
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> --- In vpFREE@yahoogroups.com, David Silvus <djsilvus@> wrote:
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> > And sometimes it refuses to see patterns based on preconceived assumptions.
> > http://www.livescience.com/animals/pigeons-monty-hall-problem-100304.html
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> So, in the Monty Hall problem, can we perceive a pattern in the remaining two doors? When 1 and 2, more likely to be 1, when 2 and 3 more likely to be 2, etc.? I don't think the pigeons did - they learned to make a choice based on probability, not patterns, and I didn't see evidence that they then chose the correct door with any edge.
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> --- In vpFREE@yahoogroups.com, "nightoftheiguana2000" <nightoftheiguana2000@> wrote:
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> >American Coin scandal, the machines would still be operating today.
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> American Coin involved gaffed machines - no randomness there.
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> >even coin flips are up to dispute:
> >http://www.npr.org/templates/story/story.php?storyId=1697475
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> Diaconis' work proved that if we flip a coin, with the same initial conditions, we can predict, nay, control, the outcome - similar to placing a coin in our palm and turning our palm over - the coin still "flipped" - with predictable results - until we lose muscular control. This is not randomness.
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> >http://en.wikipedia.org/wiki/Hardware_random_number_generator
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> Do certain hardware (and pseudo) random number generators decay? Yes, pseudos will even repeat. Do they all? I don't know - counting neutrino arrival rates, e.g., may not. Even with decay, and absent of knowledge of initial conditions, I say such devices produce sequences that are random as far as we can tell. Do we think that we can track past output from such devices and discern a future pattern? I don't.
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> I think that's Frank's question - 3 reds in a row so now bet black (or red)?, trip 6s in a row so now hold a single 6?
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