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# Gaming Guru

### Hand Wringing Hands in 9/6 Jacks

12 November 2010

9/6 Jacks or Better Video Poker is, for the most part, a reasonable game.  High cards are better than low cards; high pairs are better than low pairs, and so on.  While it is true that one needs to study a strategy table to establish a hierarchy of hands, there are almost no real surprises.  When dealt a hand with three high cards that include an Ace, it is a bit of a surprise that one should toss the Ace and hold only the other two high cards, but it is not a play that one would wring one's hands about.  There are, however, a couple of hands that most players find hard to play correctly.  Let me show you.

If you are dealt a pat flush that contains four to a Royal, the correct play is to hold the four-card Royal and discard the low suited card.  What?  Bust up a made hand?  That's right.  Notice that the expected return on the made hand is 30 units (I am assuming that 5 units are bet).  Let's look at the expected return for the four-card Royal.

I am going to assume that the four-card Royal is the weakest one possible, namely that it contains both the Ace and Ten (thereby precluding a possible straight flush).  Here is the table one uses to calculate the expected return:

 Event Freq. Pays Product No Win 27 0 0 Hi Pair 9 5 45 2 Pair 0 10 0 Trips 0 15 0 Straight 3 20 60 Flush 7 30 210 Full Hse 0 45 0 Quads 0 125 0 Strt. Flsh 0 250 0 Royal 1 4000 4000 Totals - 47 -- 4315

To convert the frequencies to probabilities one would divide each by 47.  The same thing is accomplished by dividing the final sum of 4315 by 47.  The result is 91.8085.

The expected return by drawing to the four-card Royal is over three times the expected return by just holding the pat flush.  Of course, most of this return results from the Royal Flush.  Nevertheless, the chance of drawing this a Royal is 2.1277%.  Not bad!

The other hand-wringing hand is a four-card flush that contains a three-card Royal.  Most of the time the correct play, just as in the above example, is to toss the suited low card along with the unsuited card.  A four-card flush is a good hand in that there are 9 ways to make the 5 flush, approximately a 19.15% chance of hitting it.  Nevertheless, as we will see, the expected return is better if we toss the low suited card.  As above I am going to assume the worst three-card Royal.  Suppose our hand is AH, QH, TH, 5H, 8S.  If we hold the four-card flush and draw one card, the table looks like this:

 Event Freq. Pays Product No Win 32 0 0 Hi Pair 6 5 30 2 Pair 0 10 0 Trips 0 15 0 Straight 0 20 0 Flush 9 30 270 Full Hse 0 45 0 Quads 0 125 0 Strt. Flsh 0 250 0 Royal 0 4000 0 Totals - 47 -- 300

As before, dividing 300 by 47 we obtain the expected return by holding the four flush; it is 6.38248.  On the other hand if we draw two cards to the three-card Royal, the table looks like this:

 Event Freq. Pays Product No Win 754 0 0 Hi Pair 240 5 1200 2 Pair 27 10 270 Trips 9 15 135 Straight 15 20 300 Flush 35 30 1050 Full Hse 0 45 0 Quads 0 125 0 Strt. Flsh 0 250 250 Royal 1 4000 4000 Totals - 1081 -- 6955

Here we once again divide the total by 1081; the result is 6.43386.  This is approximately five cents better than the previous expected return.

"But wait," you say.  "The chances of hitting that Royal are 1 in 1081 or approximately 0.0925%.  I think I would be willing to forego the nickel for a decent chance to win 30 units rather than taking such a long shot."

Believe it or not I have some sympathy for that position.  If you are just a casual player who gets to Las Vegas once or twice a year and plays Video Poker for an hour or two, you'll probably be happier drawing one to the flush in this case.  It is a close call.  In fact if that 8S were any high spade (or diamond or club) other than the Ace or Queen, the correct play would be to draw one to the flush.  If you play regularly, however, in the long run you'll be better off if you play the correct strategy, that is, play for the highest expected return.

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