Abstract

This study examines the probability and the expected number of hands required to "break the bank" in the 1985 Game and Watch title Blackjack, a game ending when a player’s total money either reaches zero or exceeds $9,999. This analysis is centered on the game’s unique mechanics and restrictions, particularly noting the absence of options such as splitting and insurance, which are commonly found in standard Blackjack games. The game is structured as a 1v1 matchup against the dealer, starting with a $500 balance and allowing a maximum bet of $100, which can be doubled. Notable rules include a single deck that is reshuffled after at least 12 cards are drawn, the dealer standing on soft 17, and special payouts where ties result in no change to the player's wallet except in the case of both player and dealer hitting Blackjack, where the player wins but receives only half the bet.

This Answers seeks to model the game as a random walk, considering each hand as an independent trial with associated probabilities of winning, losing, or tying, based on simplified Blackjack basic strategies. The theoretical framework employed considers the gambler's ruin problem to estimate the likelihood of achieving a wallet amount of $9,999 before bankruptcy, alongside calculations to estimate the expected number of hands needed under these conditions.

APPROACH

  1. Basic Strategy: Since splitting is not allowed in this version of Blackjack, we can focus on basic strategy without considering split hands. Basic strategy charts are available for games with similar rules, such as standing on soft 17 and doubling down allowed on any two cards.
  2. Deck Composition: The probability of drawing specific cards from the deck affects the player's and dealer's chances of winning each hand. With only one deck in play, the composition changes with each hand, especially since the deck is reshuffled after a certain number of cards have been drawn.
  3. Simulation: Given the complexity of the game and the various factors involved, simulating multiple rounds of play can provide insights into the likelihood of reaching the maximum wallet amount without losing. By running simulations, we can track the player's bankroll over multiple hands and determine the frequency with which the player reaches or exceeds the maximum amount.
  4. Monte Carlo Method: Using the Monte Carlo method, we can simulate thousands or even millions of rounds of Blackjack based on the game rules and basic strategy. This approach involves randomly generating hands and outcomes according to the probabilities associated with each action (hit, stand, double down) and comparing the results to the player's bankroll.
  5. Analysis: Once we have the results of the simulations, we can analyze the data to determine the probability of "breaking the bank" and estimate the average number of hands it takes for a player to reach the maximum wallet amount.

First Code: Runned 10000

import numpy as np

def simulate_blackjack(starting_funds, win_probability, target, bet_size):
    funds = starting_funds
    hands_played = 0
    
    while funds > 0 and funds < target:
        if np.random.rand() <= win_probability:
            funds += bet_size
        else:
            funds -= bet_size
        hands_played += 1
        
        # Adjust bet size if it would result in negative funds
        if funds < bet_size:
            bet_size = funds
    
    return funds >= target, hands_played

# Simulation parameters
win_probability = 0.49  # Assuming a win rate close to neutral given the game rules
starting_funds = 500
target = 9999
bet_size = 100
trials = 10000

results = [simulate_blackjack(starting_funds, win_probability, target, bet_size) for _ in range(trials)]
successes = sum(result[0] for result in results)
average_hands = sum(result[1] for result in results) / trials

print(f"Probability of breaking the bank: {successes / trials:.2%}")
print(f"Average number of hands played: {average_hands:.1f}")

Results: 

Probability of breaking the bank: **0.46%**
Average number of hands played: **226.7**

SECOND CODE (To achieve more accuracy and consistency)

  1. Maintain Bet Limits: Keep the bet within the allowed maximum of $100, doubling to $200 when favorable.
  2. Improved Decision Logic: Consider when doubling might be beneficial. However, a precise calculation would require a more sophisticated model of card probabilities, which could be complex given frequent reshuffles.
  3. Increase Trials for Robustness: Running more trials can improve the statistical robustness of our results.
import numpy as np

def simulate_blackjack(starting_funds, win_probability, target):
    funds = starting_funds
    hands_played = 0
    bet_size = 100  # Standard bet size

    while funds > 0 and funds < target:
        if funds < bet_size:
            bet_size = funds  # Adjust bet size if funds are low
        
        # Decision to double down (simplified assumption: double down on favorable random chance)
        if np.random.rand() < 0.1 and funds >= bet_size * 2:
            current_bet = bet_size * 2
        else:
            current_bet = bet_size
        
        if np.random.rand() <= win_probability:
            funds += current_bet
        else:
            funds -= current_bet
        hands_played += 1
    
    return funds >= target, hands_played

# Simulation parameters
win_probability = 0.49  # A rough estimate for a basic strategy outcome
starting_funds = 500
target = 9999
trials = 10000

results = [simulate_blackjack(starting_funds, win_probability, target) for _ in range(trials)]
successes = sum(result[0] for result in results)
average_hands = sum(result[1] for result in results) / trials

success_rate = successes / trials
success_rate, average_hands

Results: 

Probability of breaking the bank:  0.58%
Average number of hands played: 201 hands