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# Tossing Aces in 9/6 Jacks

11 March 2012

One of the counterintuitive plays in 9/6 Jacks or Better video poker occurs when one is dealt three high cards and one of them is an Ace. The correct play is to just hold the two non-Aces and discard the Ace. When I tell people this they frequently ask me to explain why this makes sense. Of course the real answer is that the expected return to the player is better than the expected return when holding all three high cards. This, however, is not the answer most people want; they want to intuitively understand why tossing the Ace is the better play.

In an attempt to answer this I want to look at a specific example. Suppose you are dealt the following hand:

AH, QS, JD, 7H, 5C

Doesn't holding the AQJ increase the chances of getting a high pair? Yes, but not by much. From the following table one can deduce that the probability of getting a high pair increases from 30.31% when holding the QJ to 32.19% when holding the AQJ. The table also shows some other factors.

 Hand Payout QJ Freq. Product AQJ Freq. Product Garbage 0 10,113 0 681 0 High Pair 1 4,914 4,914 348 348 2 Pair 2 711 1,422 27 54 Trips 3 281 843 9 27 Straight 4 176 704 16 64 Flush 6 0 0 0 0 Full 9 18 162 0 0 4 of Kind 25 2 50 0 0 SF 125 0 0 0 0 Royal 800* 0 0 0 0 Totals --- 16,215 8,095 1,081 493

*Assuming 5 coins are played.

Notice that by holding the QJ there is a higher percentage of high pairs than when holding AQJ (4.38% to 2.49%) and there are no Full Houses or Quads when holding AQJ. Also the only type of straight when holding AQJ is TJQKA, whereas when holding the QJ there are straights of the form 89TJQ, 9TJQK, and TJQKA. In brief, holding only the QJ produces more types of hands than when holding AQJ.

Assuming that 5 coins are played when holding QJ, we can calculate the coin return to the player to be 5 x (8095/16215) or 2.496 coins. For the AQJ the return to the player is 5 x (493/1081) or 2.280. Notice that both of these are losing situations, but by holding the QJ one loses less. I hope this explains why holding the QJ is the better play. Similar results hold for AKQ and AKJ 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