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# Gaming Guru

### Three Card Flushes

6 May 2012

When playing 9/6 Jacks or better Video Poker it is tempting to hold a three-card Flush. After all there are still ten suited cards in the deck so this seems like a reasonable play. Unless there is a possible Straight Flush draw, however, it is the wrong play. Even some hands that are possible Straight Flushes are not always worth holding; these have to be analyzed separately using proper 9/6 Jacks strategy. So let me show you by example that you should not hold three card Flushes in this game.

Consider the hand KC, JC, 7C, 8D, 4H. The following chart shows that holding the KJ is better than holding the KJ7.

 Hand Pay KJ Prod KJ7 Prod Zip 0 9,935 0 766 0 J or B 1 5,022 5,022 240 240 2 Pair 2 711 1,422 27 54 3K 3 281 843 9 27 Str. 4 126 504 0 0 Flsh. 6 118 708 45 270 Full 9 18 162 0 0 4 Kind 25 2 50 0 0 SF 50 1 50 0 0 Roy *800 1 800 0 0 Totals --- 16,215 9,561 1,081 541

*Assumes 5 coins played

The third and fifth columns show the hand frequencies and the fourth and sixth columns show the number of coins returned to the player. The amount returned to the player per coin played is 9561/16215, or 0.5896, for the KJ and 541/1081, or 0.5467 for the KJ7.

Now one might reason that if there were no high cards in the flush then holding the three flush is better than getting a whole new hand. The following table shows that this isn't so. Consider the hand TC, 7C, 4C, 6D, 9H.

 Hand Pay None Prod T74 Prod Zip 0 1,178,068 0 976 0 J or B 1 241,680 241.680 24 24 2 Pair 2 71,802 143,604 27 54 3K 3 31,502 94,506 9 27 Str. 4 5,296 21,184 0 0 Flsh. 6 3,103 18,618 45 270 Full 9 2,124 19,116 0 0 4 Kind 25 344 8,600 0 0 SF 50 17 850 0 0 Roy 800 3 2,400 0 0 Totals --- 1,533,939 550,558 1,081 335

Getting a whole new hand has an expected return per coin played of 550558/1533939 or 0.3584. Holding the three flush yields a return per coin played of 335/1081 or 0.3469. One can infer from this example that when dealt a three flush with one high that one should hold the high card.

So when playing Video Poker, should you always discard a three Flush? Yes, if the game is 9/6 Jacks. However if you were playing 10/7 double Bonus Video Poker you should hold some three Flushes; a strategy table for the game will tell you when. The lesson here is that you should learn to play one game well and not infer strategy decisions from it for other games. Common sense I guess. 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