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# Where Are Those Straight Flushes?

6 October 2002

I have a friend who is an avid 9/6 Jacks player. She knows how to play the game and I recently purchased Bob Dancer Presents WinPoker for her so that she could practice. In one of our conversations she remarked to me that she expected to see more Straight Flushes than actually occur and wanted my take on this. It is an interesting question.

A cursory look at the situation would suggest that Straight Flushes should occur 10 times more frequently than Royal Flushes. Cursory is right. It is true that Straight Flushes occur 10 times more frequently than Royal Flushes off the top. Remember, however, that most Royal Flushes don't occur off of the top; they are obtained by drawing to a possible Royal. Consider the hand AC, KH, QH, 4D, 2S. The correct play here is, of course, to toss the Ace, 4 and 2 and keep the suited King and Queen, a possible Royal hand. On the other hand, consider the hand AC, 9H, 8H, 4D, 2S. An analogous play here would be to toss the same three cards and keep the suited 8 and 9, a possible Straight Flush draw. Of course, that would be a foolish play but playing in that manner would certainly increase the frequency of Straight Flushes. It would also drain your resources. The point is that proper play decreases the frequency of Straight Flushes from the "off the top" number to about 5.5 times the number of Royals.

There is another point to be made here. You may be seeing fewer Straight Flushes than you should because you are playing certain poor hands incorrectly. Consider the hand 2D, 3D, 6D, 8H, 9S. If you toss this hand for an entirely new hand you are making a mistake; you should keep the double inside Straight Flush. With 5 coins played, keeping the Straight Flush has a return of 2.2155 coins whereas tossing the entire hand has a return of only 1.8016 coins. This is the lowest hand one should keep in 9/6 Jacks, but clearly it should be kept.

Here is another possible error. Suppose you are dealt AD, 2D, 3D, JH, 8S. There are two reasonable ways to play this hand. Since two high cards are better than a low double inside Straight Flush you might be tempted to keep the Ace and Jack and discard the rest. Notice though that the Straight Flush has a high card in it. Maybe keeping the Ace, 2, and 3 is the better play. In fact, keeping the Ace, 2 and 3 has an expected return of 2.7012 coins whereas keeping the Ace and Jack has an expected return of 2.2913 coins.

So, don't expect a lot of Straight Flushes in this game, but make sure you are not missing out on your share because you are misplaying these low double inside Straight Flushes.

See you next month.

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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