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The Optimal Strategy for Playing Lucky 13

Steven Berger, Daniel Conus

TL;DR

This paper develops binomial-based probabilistic models to derive optimal Lucky Range and Lucky Number strategies for Lucky 13 under two utility functions. It introduces two- and three-category question models, where the total correct answers $N$ is modeled as $N = N_S+N_G$ or $N_S+N_U+N_G$, and validates results with Monte Carlo simulations and case studies of actual contestants. It shows how risk preferences influence strategy and demonstrates that Monte Carlo validation aligns with analytic results, while also extending to a zero-category Poisson-binomial model guided by Darroch's rule. The work provides practical decision guidelines for contestants and a flexible framework for future extensions in game-show strategy.

Abstract

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

The Optimal Strategy for Playing Lucky 13

TL;DR

This paper develops binomial-based probabilistic models to derive optimal Lucky Range and Lucky Number strategies for Lucky 13 under two utility functions. It introduces two- and three-category question models, where the total correct answers is modeled as or , and validates results with Monte Carlo simulations and case studies of actual contestants. It shows how risk preferences influence strategy and demonstrates that Monte Carlo validation aligns with analytic results, while also extending to a zero-category Poisson-binomial model guided by Darroch's rule. The work provides practical decision guidelines for contestants and a flexible framework for future extensions in game-show strategy.

Abstract

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

Paper Structure

This paper contains 11 sections, 27 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Game Description
  3. Model with Two Question Categories
  4. Model with Three Question Categories
  5. Monte Carlo Simulation
  6. Case Study 1
  7. Case Study 2
  8. Simulation of Number of Correct Responses
  9. Model with Zero Question Categories
  10. Conclusion and Future Work
  11. Optimal Strategy for Selecting Lucky Range and Number (3 Category Model)

Figures (8)

  • Figure 1: 10,000 Monte Carlo Simulation Results for Contestant $Z$
  • Figure 2: Estimated Probabilities of Correct Answers for Contestant $B$
  • Figure 3: Expected Winnings for Contestant $B$
  • Figure 4: Estimated Probabilities of Correct Answers for Contestant $C$
  • Figure 5: Expected Winnings for Contestant $C$
  • ...and 3 more figures