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Soft Doubling - Part 14 April 2008
I recently received an email from reader Thomas Moore:
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:
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)
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)
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)
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.
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 firstname.lastname@example.org
This article is provided by the Frank Scoblete Network. Melissa A. Kaplan is the network's managing editor. If you would like to use this article on your website, please contact Casino City Press, the exclusive web syndication outlet for the Frank Scoblete Network. To contact Frank, please e-mail him at email@example.com.
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