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Understanding Probabilities and Equilibria in Ball Selection: Insights from Model Thinking

This text delves into the concept of probabilities in a selection process involving blue and red balls, as discussed by Scott E. Page in "Model Thinking". It explores the scenarios where selections lead to various combinations of blue and red balls, illustrating the idea that any sequence of selections (like B, R, B, B or R, B, R, B) holds equal likelihood. By emphasizing the equilibrium of probabilities in different configurations, this analysis demonstrates critical thinking about randomness and equal chances in decision-making processes.

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Understanding Probabilities and Equilibria in Ball Selection: Insights from Model Thinking

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Presentation Transcript

  1. Model Thinking Scott E Page

  2. Two “Proofs”

  3. Polya U = {1 Blue, 1 Red} Select and return Add a new ball that is the same color as the ball selected

  4. Polya in Action

  5. Result 2: Any history of B blue and R red balls is equally likely

  6. P(RBBB) = P(BBBR) =

  7. P(RBRB) = P(BBRR) =

  8. Result 1: Any probability of red balls is an equilibrium and equally likely.

  9. P(BBBB) = P(BBBR) =

  10. P(50B) = P(49B 1R) = P(47B 3R) =

  11. Result 1: Any probability of red balls is an equilibrium and equally likely.

  12. Result 2: Any history of B blue and R red balls is equally likely

  13. Model Thinking Scott E Page

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