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Best of Donald Catlin

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Soft Doubling - Part 2

11 May 2008

Last month, in answer to my reader Thomas Moore inquiring about soft doubling, I indicated how to calculate dealer's final probabilities when the up card is a 6. In that article I gave a table listing the probabilities for dealer's final totals of 17 through 26. Since the hands 22 through 26 are all busted hands, we can add all of those probabilities and simply list it as bust. Here is the resulting table that we will use.

Hand

Probability

17

0.1148

18

0.1148

19

0.1148

20

0.1103

21

0.1057

Bust

0.4395

Recall that we are addressing the situation where the player's hand is a soft 18 and he is facing a dealer's 6. Assume that we have already determined that the player should stand on a hard 12 through 21 when facing a 6. Also let us assume that, using calculations such as those below, we know that we should stand on a soft 19, 20, and 21 when the hand consists of three or more cards. What about the soft 18?

If we stand on it we get the following table for calculating expected return:

Deal. Tot.

Payout

Prob.

Product

17

+1

0.1148

+0.1148

18

0

0.1148

0

19

-1

0.1148

-0.1148

20

-1

0.1103

-0.1103

21

-1

0.1057

-0.1057

Bust

+1

0.4395

+0.4395

Totals

--

0.9999

+0.2235

So, if we stand on our Ace-Seven versus the 6 we have a positive expected return of 0.2235. What if we hit the hand? We know that if we do hit the hand we will only hit it once. Let's look into this situation.

Suppose we hit any soft 18 and draw an Ace. Our expected return in this situation can be calculated using the following table.

Deal. Tot.

Payout

Prob.

Product

17

+1

0.1148

+0.1148

18

+1

0.1148

+0.1148

19

0

0.1148

0

20

-1

0.1103

-0.1103

21

-1

0.1057

-0.1057

Bust

+1

0.4395

+0.4395

Totals

--

0.9999

+0.4531

Let me do one more calculation so you get the idea. If we draw a 4 through 8, we end up with a stiff, a 12 through 16. In this case the only way to win is if the dealer busts. Hence:

Deal. Tot.

Payout

Prob.

Product

17

-1

0.1148

-0.1148

18

-1

0.1148

0.1148

19

-1

0.1148

-0.1148

20

-1

0.1103

-0.1103

21

-1

0.1057

-0.1057

Bust

+1

0.4395

+0.4395

Totals

--

0.9999

-0.1209

I think you can check the rest of the expected returns for the various draws yourself using the above ideas. This results in the following calculation for overall expected return when drawing.

Draw Card

Prob. X 13

Exp. Return

Product

Ace

1

0.4531

+0.4531

2

1

0.6782

+0.6782

3

1

0.8942

+0.8942

4 thru 8

5

-0.1209

-0.6045

9

1

-0.0061

-0.0061

10

4

0.2235

+0.8940

Totals

13

----

+2.3089

Because I used 13 times the card's probability in the calculation (to reduce round off error) the 2.3089 is 13 times larger than the correct expected return. Dividing by 13 we obtain 0.1776. This is also a positive expected return.

Since our expected return by standing is 0.2235 and our expected return by hitting is 0.1776, the conclusion is that if our soft total of 18 contains three or more cards we should stand. However, if our soft 18 consists of two cards and we have the option of doubling, our expected return is 2 x 0.1776 or 0.3552 and this is a much better return than standing.

So, Thomas, when you have a soft 18 versus the 6, you should stand on it unless it is an Ace-Seven hand, then you should double it. I hope this answers your question to me. If you get any guff from those ploppies that you mentioned in your email to me about this strategy, refer them to these two articles. I guarantee they won't understand them. Ignorance might be bliss (or obnoxious) but it's not profitable. See you all next month.


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