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Best of Donald Catlin

Gaming Guru

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Sixteen Versus Seven

1 April 2006

I often read the column of my friend and colleague John Brokopp, who also writes for this site. John recently had a series of columns devoted to interviews with a successful Blackjack player identified only as Mr. Aces. In John's February 8, 2006 article he asked Mr. Aces if he used Basic Strategy all the time. The answer was that it is important to use Basic Strategy all the time. He went on to say that on rare occasions he does deviate from it and that "quite frankly it is one of my downfalls." The specific example he gave is the situation wherein the player has a total of 15 or 16 and the dealer shows a 7. Here are his exact words.

"...I have a tendency to hesitate, especially if I have a large bet on the table. Basic strategy tells you to hit the hand, but when I see that dealer's seven I think that if there is anything but a ten or ace in the hole he's going to have to hit the hand."

It is interesting to me that this guy, who obviously knows better, would admit to hesitating in this situation and perhaps even intentionally misplaying the hand. It is also understandable. After all, the seven is not a very good card for the dealer and one's gut reaction is to sit back and let the dealer draw, this being the most probable scenario. Gut reaction or not, standing is the wrong play and I'd like to show you why.

I'm going to do this analysis two ways. One is anecdotal and the other is to show you the exact numbers involved. I will use an infinite deck analysis for simplicity. As Peter Griffin points out in his book Theory of Blackjack, an infinite deck is not infinite at all but rather is a deck used in a game where one draws with replacement. This means that the probabilities don't change as the game progresses. This is a good approximation for multiple-deck games. The more decks the less the probabilities are affected by the removal of a card and thus the reason for the infinite deck terminology.

The probability of a ten is 4/13 and the probability of any other card is 1/13. Thus there is a 5/13 probability that by standing you will be beaten. But there is also a 3/13 probability that the dealer will draw a two, three, or four, which would result in a good drawing hand for the dealer -- that is, a nine, ten or eleven. This means that 8/13 of the draws are favorable for the dealer, only 5/13 of them are not. Of course, some of the good draws for the dealer may result in busts but some of the bad draws may result in good hands; it takes a more refined analysis to sort these out. Thus the above argument is not definitive but it sure is enough to make me doubt the wisdom of standing.

A precise argument would have to look at all of the possible situations and look at the expected returns by either standing or hitting and compare the two. I am going to assume an infinite deck game with the dealer hitting the soft seventeen (which is actually good for the player in the situation we're discussing). I wrote a program that calculates these expected returns and here are the results. If the player holds a sixteen versus the dealer's seven, the player has an expected return of approximately -0.4754 units if he stands and an expected return of -0.4148 if he hits. The difference is 0.0606. This means that the player is losing about six cents per dollar in this situation if he stands. If the player hits on 15 his expected return is -0.3698 and so standing in this situation costs the player about ten and a half cents per dollar.

So, grit your teeth and hit that sixteen versus the seven; in the long run you'll be glad you did. See you next month.

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