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Best of Donald Catlin

Gaming Guru

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Soft Doubling - Part 1

4 April 2008

I recently received an email from reader Thomas Moore:

It is hard for me to understand how doubling on an ace - x a dealer's 5 or 6 up card is the correct play, where x is a deuce through seven. Please help me understand this basic strategy play.

Thanks.

I sent Thomas the following reply: This is not easy to explain but I'll give it a try in an upcoming article. Thomas replied:

Thanks, I understand the difficulty. I do the double downs as recommended but this throws other players into a tither on how anyone would double down on an Ace 5 versus a 5 or 6. Since I'm a low roller, the normal comment is how they let an unknowledgeable player betting the minimum mess up my big bets.

By the way, as a Basic Strategy player I try not to sit at third base just to try and reduce such comments. This makes the hitting of Ace 7 (soft 18) against 9, 10, or Ace less noticed too.

Well, Thomas, it sounds as if you've encountered your share of ploppies (as Frank calls them). Don't be intimidated; your Basic Strategy play is correct.

In this article and the following I am going to address the specific situation of Ace 7 versus the dealer's 6. To start the analysis we need to know the probabilities of each of the dealer's final totals when he shows a 6 and that is what I'll address here.

I am going to use an infinite deck analysis, which really means sampling with replacement. This is a good approximation for multiple decks and makes the calculations easy. Under this assumption the probability p(x) of drawing x is 1/13 when x is Ace through 9 and p(10) = 4/13.

It is convenient to separate Ace and non Ace hands. Let h(x) represent the probability of the dealer having a non Ace total of x. At the start h(6) = 1. Since we cannot obtain a non Ace total of 7, h(7) = 0. And...

h(8) = p(2)h(6)
h(9) = p(2)h(7) + p(3)h(6) [notice the first term is zero]
h(10) = p(2)h(8) + p(3)h(7) + p(4)h(6)
h(11) = p(2)h(9) +p(3)h(8) + p(4)h(7) + p(5)h(6)
h(12) = p(2)h(10) + p(3)h(9) + p(4)h(8) + p(5)h(7) + p(6)h(6)

I think you can see the pattern here. Recursively we can calculate all of the non Ace totals from 6 to 26, being careful to use the dealer's hit/stand rules when we get to totals above 16.

The Ace hands are a bit trickier. Starting with a 6 the smallest soft total is A 6 which is a soft 17. Let's let s(x) stand for a total of x when the hand has one or more aces. Then...

s(17) = p(1)h(6)
s(18) = p(1)s(17) + p(1)h(7) [dealer hits soft 17]
s(19) = p(2)s(17) + p(1)h(8)
s(20) = p(3)s(17) + p(1)h(9)

For totals above 21, we use hard hit stand rules, that is, for totals 22 through 26 (hard 12 through 16) we hit, for totals 27 through 31 (hard 17 through 31) we stand, and totals 32 through 36 are busted hands.

Assuming all of these calculations have been carried out correctly we calculate the dealer's probabilities as follows:

d(17) = h(17) + s(27)
d(t) = h(t) + s(t) + s(t +10) for t 18 through 21
d(t) = h(t) + s(t+10) for t 22 through 26

Naturally all of this is a great deal of work using pencil and paper, but the patterns involved are so nice that with some effort one can write a computer program to carry out the calculations. That is exactly what I did. If any of you reading this article can program, you might this using my outline above. If you do, make sure that you check to see that your dealer probabilities add up to 1.

Here are the results of my program rounded to 4 places.

  Dealer Total    Probability  
170.1148
180.1148
190.1148
200.1103
210.1057
220.0910
230.0846
240.0778
250.0704
260.1157

Next month I'll show you how we can use the above numbers to answer Thomas's question. See you then.


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

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