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# Gaming Guru

### Weak Three-Card Straight Flushes in 9/6 Jacks

1 November 2009

Let me begin by saying what I mean by a weak three-card straight flush. By this I mean any three-card straight flush that has one high card and is double inside or has no high cards and is single inside. A very weak three-card straight flush is one that is double inside and has no high cards. This latter is the weakest hand that one keeps before drawing a whole new hand and won't concern us in this article.

If you look at a 9/6 strategy table you will see that a weak straight flush is better than an unsuited KQJ hand. You'll also note that either an AT hand with two high cards or a KQJ9 hand is better than a weak three-card straight flush. This is usually true but not always. Let's look at this.

Consider a hand such as AD, QH, JS, TH, 8H. The hand QT8 is not the preferred choice here; the correct choice is to hold the AQJT just as the table suggests. On the other hand, consider the hand AD, QS, JH, TH, 7H. If you look at the AQJT8 example, you'll note that holding the QT8 removes a jack from the possible cards to fill a straight. This is called straight interference and is enough to make QT8 the poorer choice. In the second hand, however the JT7 does not have any straight interference and this fact makes it a better choice than the AQJT hand. Here are the details. For the JT7 hand we have the following:

 Final Freq Pay Prod Zip 859 0 0 JOB 126 5 630 2 Pr 27 10 270 3 K 9 15 135 Str. 15 20 300 FL 44 30 1320 FH 0 45 0 4K 0 125 0 SF 1 250 250 Roy 0 4000 0 Totals 1081 -- 2905

If we divide the total 2905 by the number of hands, 1081, the result is 2.6873 and represents the average return one can expect when playing the hand this way. On the other hand for the AQJT hand we have this table:

 Final Freq. Pay Prod. Zip 34 0 0 JOB 9 5 45 2 Pr. 0 10 0 3 K 0 15 0 Str. 4 20 80 FL 0 30 0 FH 0 45 0 4 K 0 125 0 SF 0 250 0 Roy 0 4000 0 Totals 47 --- 125

In this case if we divide 125 by 47 the result is 2.6596, the average return we can expect when playing the hand this way. As you can see the QT7 hand is better than the AQJT hand by about 2.77 cents.

In general if an AT hand contains two high cards and a weak three-card straight flush, then the straight flush is the better choice if there is no straight interference. The same is true of a KQJ9 hand. See you 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