Chuck A. Luck Wagers a Buck:
A Case Study in Probabilistic Reasoning and the Gambler's Ruin

Christopher M. Rump
Department of Industrial Engineering
University at Buffalo, State University of New York

PART II - Going for Broke

Chuck A. Luck has brought $2 to the casino. He will play Chuck-a-Luck (wagering $1 on each play) until he has $4 or until he goes broke, whichever comes first.


  1. Formulate a discrete-time Markov chain for this gambler's ruin problem by defining an appropriate state space. Write the transition probability matrix for this chain. Does the Markov assumption hold in this case? Explain.

  2. What is the probability that Chuck goes home broke?

  3. On average, how many chucks does Chuck chuck, i.e., how long does the game last?

  4. Suppose Chuck considers a strategy of wagering $2 instead of $1 on each play. How would these answers change? Which strategy should Chuck employ if he wants to minimize his probability of going broke? If he wants to maximize the number of plays? Explain.

  5. Play repeated rounds of this Chuck-a-Luck game against a friend. Choose one player to act as the "chucker" and one person to play as the "house." Both the "chucker" and the "house" should start with $2 each. Here, each game consists of either you or your friend going broke. Be sure to keep track of how long each game lasts and who won. Do your "simulated" results match the expected theoretical results?