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    Bertrand's chord paradox shows that three geometrically v... — Carmelics
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    Supports→The classical interpretation of probability yields implausible and contradictory results when applied generally.

    Bertrand's chord paradox shows that three geometrically valid but non-equivalent equipossibility partitions yield probabilities of 1/3, 1/4, and 1/2 for the same event, proving internal inconsistency.

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    Reasons For

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    Reason for
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    • 1.The three methods produce mathematically correct results under their own definitions, yet contradict each other, demonstrating that probability requires external assumptions.
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    • 2.If equipossibility were a self-evident principle, identical geometric objects couldn't yield incompatible probability assignments without logical contradiction.
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    • 3.The paradox reveals that 'geometric validity' alone cannot determine probability—something non-geometric must specify which partition reflects reality.
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    Reasons Against

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    • 1.The three methods solve different problems (random endpoints vs. random radii vs. random midpoints), so disagreement reflects different questions, not internal inconsistency.
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    • 2.Inconsistency requires that the same setup yield contradictory probabilities; but different randomization procedures are not 'the same event' mathematically.
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    • 3.The paradox shows probability depends on *how* we generate randomness, not that equipossibility is incoherent—physics, not logic, determines the correct method.
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    Key Terms

    Bertrand (Bertrand Russell)(The paradox is named after him)
    A famous British philosopher and mathematician (1872-1970) who created puzzles and paradoxes to explore how we think about probability and logic.
    Equipossibility(The paradox shows three different ways to partition outcomes that all seem equally valid)
    The assumption that all possible outcomes or ways of doing something are equally likely to occur.
    Internal inconsistency(The paradox shows that the concept of equipossibility contains a flaw)
    A contradiction or logical problem within a system itself—when a single theory or method produces conflicting answers to the same question.
    paradox(R. M. Sainsbury's definition, presented as a target of criticism)
    An apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises
    partition(Used to describe the structure of Pascal's decision matrix)
    A division of the hypothesis space into mutually exclusive and jointly exhaustive alternatives, such as 'God exists' and 'God does not exist' by the law of excluded middle.
    probability(as used in mathematics and logic)
    A number between 0 and 1 that describes how likely something is to happen; 0.5 means 50% chance, 1 means certain, 0 means impossible.

    Connections

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    Natural Theology1 linked

    Related

    If equipossibility were a self-evident principle, identical geometric objects co...Inconsistency requires that the same setup yield contradictory probabilities; bu...The classical interpretation of probability yields implausible and contradictory...The paradox reveals that 'geometric validity' alone cannot determine probability...

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    The paradox shows probability depends on *how* we generate randomness, not that ...The three methods produce mathematically correct results under their own definit...The three methods solve different problems (random endpoints vs. random radii vs...