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# Gaming Guru

### A Royal Dilemma

4 October 2009

A choice that (unfortunately) one seldom has when playing 9/6 Jacks or better Video Poker is that of choosing between a three-card Royal Flush and a four-card Flush.  All Basic strategy tables for 9/6 Jacks list the three-card Royal as being better than the four-card Flush.  For the most part this is good advice.  However, there is one case where the opposite is true.  A hand that contains a suited Ace and ten along with another suited high card is a weak three-card Royal since it is a double inside situation and thus reduces the straight possibilities.  This in and of itself does not negate the Basic strategy rule; it depends upon the nature of the fifth card in the hand.  Let me illustrate with an example.

Suppose you are dealt AH, QH, TH, 9H, X.  If X is an unsuited low card, say a three of clubs, the correct play would still be to hold the AQT.  However if X is an unsuited King or Jack then this reduces the number of straight, 2 pair, and high pair hands on the draw and is enough to render the four-card Flush the better hand.  What if X is an unsuited ten?  Let's look at that situation in detail.

Suppose we are dealt the AH, QH, TH, 9H, TC.  Discarding the 9H and TC means that we have C(47, 2) or 1081 possible draws.  The following table shows the results.

 Result Freq. Pays Product Zip 756 0 0 High Pair 246 5 1230 2 Pair 21 10 210 3 Kind 7 15 105 Straight 15 20 300 Flush 35 30 1050 Full House 0 45 0 4 Kind 0 125 0 Str. Flsh. 0 250 0 Royal 1 4000 4000 Totals 1081 -- 6859

Dividing the total by 1081 we obtain the expected return of 6.3784.  On the other hand if we only discard the TC there are 47 possible draws and we have the following table.

 Result Freq. Pays Product Zip 32 0 0 High Pair 6 5 30 2 Pair 0 10 0 3 Kind 0 15 0 Straight 0 20 0 Flush 9 30 270 Full House 0 45 0 4 Kind 0 125 0 Str. Flsh. 0 250 0 Royal 0 4000 0 Totals 47 -- 300

Dividing the total by 47 we obtain the expected return in this situation as 6.3830.  The expected returns here are really close but holding the four-card Flush is the better choice.

I should add that if X were either an Ace or a Queen then you would have a high pair and that is better than either the three-card Royal or the four-card Flush.  If you remember this example you should easily play these Ace, high, ten hands.  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