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# A Blackjack Ruin Problem

1 February 2008

Dear Don,

I am hopeless at mathematics but I understand at Blackjack I have a 43% chance of winning and a 49% chance of losing - that is a difference of 5% advantage to the casino.

I am guessing that perfect basic strategy, double downs and splits, are responsible for bringing the advantage to 0.5% - so now the casino does not have a 5% advantage - so what % of winning do or losing does the casino and myself have?

The other, more important question is - in say 75 hands (an hour of play) what chances do I have of winning \$300 with \$100 bets - I ask this because I know the number of hands I win and lose will not change, and that only splits and double downs will make the difference isn't it true that playing fewer hands (an hour or less) is to my advantage?

What are the odds of my winning \$300 with \$100 bets every hour (assuming a new show (shoe??) every time I win \$300?

Thanks,
Jack

I wrote back to Jack and pointed out that his first question is not what is important to the player. Rather it is the house edge and that is readily available in the many Blackjack books on the market.

I also asked Jack where he got the 43%/48% figures. I consulted five different Blackjack references (including Griffin) and nowhere could I find any reference to these numbers. This is not surprising since win/loss percentages will depend upon the particular set of rules one is facing. Anyway, Jack did not specify his reference and simply said that these are facts.

Jack's second question is interesting but not, I believe, well posed. Presumably, Jack is willing to risk a certain number of dollars to win \$300 but doesn't say what that amount is. I pointed this out to him but he never did specify an amount in his answer to me. So, I am going to take a guess and make his initial stake \$1000. He either plays till he reaches or exceeds \$1300 or he goes broke. This is a classical gambler's ruin problem; however with the doubles, splits, and 3:2 payoff on Blackjack cannot be handled using the standard mathematical techniques.

I wrote a computer simulation wherein the player starts with 20 units, bets 2 units at a time, and plays until he reaches or exceed 26 units or has less than 2 units left. The rules are the ones usually found today, namely, double on first two cards, double after split, split up to three times for four hands, Aces split once and receive one card, and dealer hits soft seventeen. I used a four-deck shoe.

I ran 150 million hands, which resulted in 5,970,218 rounds of either winning 6 units or losing 20. This is a bit over 25 hands per round on average. Of these, 1,574,090 were ruin situations and 4,396,128 were wins, which produces odds a bit better than 11 to 4 of reaching or exceeding \$300 when starting with \$1000. I should point out that 150 million hands are not really enough to produce tight figures so the 11:4 should be viewed as a ball park estimate.

Jack, I hope this satisfies your curiosity. See you all next month.

Don Catlin can be reached at 711cat@comcast.net

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers