Did The Professor Screw Up?

Which professor? Yes, I'm afraid the question refers to me. Here's the story.

In October I flew to Las Vegas to attend the World Gaming Congress & Expo. The show was bigger than ever and there were lots of new games on display. I'll be writing about some of them in future articles, I'm sure. While in Las Vegas I stayed at the Santa Fe, out at the end of Rancho Drive in North Las Vegas. This has been a favorite of mine for many years. In my August 2000 column, I reported that the Santa Fe had been sold to the Station Casinos chain and I expressed some concern about the changes that might take place there. Well, there's plenty of change. Construction was going on at a rapid pace while I was there, but I have not been back since to see the results. I do know that all of the original specialty restaurants are gone (including Suzettes), but I have not seen the replacements. The new facility will be called Santa Fe Station. I am hoping that the great Video Poker for which the Santa Fe was noted will be left intact. Their Blackjack seemed okay when I was there. I have since had some disappointing interaction with a casino host (people should call back when they say they will), so I am antsy about this new operation but we'll see. I'll keep you posted in future articles.

One of the great Video Poker games offered by the old Santa Fe was a 25-cent double progressive 9/6 Jacks or Better game. By double progressive I mean the following. There are two progressive totals running simultaneously; one is the current payoff on a Royal and the other is "waiting in the wings," so to speak. When the Royal is hit, the second progressive immediately becomes the next Royal payoff and a new secondary progressive is started at $1000. This means that the Royal payoff always starts at an amount greater than $1000. The game becomes even at around $1275, and I have frequently seen the Royal payout above this amount.

When I visited the Santa Fe in October, the double progressive 9/6 game was still operating; I hope it will continue to do so. Nevertheless, it was here that I might have screwed up.

Now I am not a "dyed in the wool" Video Poker player. In fact, I'm not a dyed in the wool gambler. I'm probably just like you. I get to Las Vegas three or four times a year and that is when I gamble. That's right, I'm a casual player. I generally play Blackjack since I seem to have all of the stuff I need to play tucked away in my head. This is not surprising since most of the plays in Blackjack, if one reflects on them, are intuitively clear; there are only a few that need to be memorized. Not so Video Poker. Video Poker is replete with all sorts of exceptions, penalty cards, close calls, you name it. (The worst is probably 10/7 Double Bonus. Bob Dancer writes in the introduction to his pamphlet on the game: "This is an ugly game.") Nevertheless, after an hour or two at the Blackjack table making $25-$50 bets, it feels good to unwind and play some Video Poker at $1.25 a pop and the 9/6 double progressive game suits me just fine.

Since I play Video Poker 3 or 4 times a year, it does me no good to memorize all of the nuances of the game because I forget them in between times. Maybe you're the same way. Now, I do know that a low double inside straight flush is the lowest hand one should keep in 9/6 Jacks. Why do I know this? Because knowing this tells me when to throw in all five cards, a situation that is not infrequent. I know that a low pair is inferior to a 3-card Royal and a 4-card Flush but is superior to any 4-card Straight. A high pair is better than any of these. Why do I know this? Because a pair will occur on average once in every 2.36 hands, so one ought to know about pairs. Well, you get the idea. You should know the frequent plays and they'll stick with you because you get them so often. The rest? I use a crib sheet. That's right, I have a small set of cards that I use when I play that have all of the unusual hands listed. If I get a hand that I don't remember, I quickly look it up and go on. The longer I play, the less I use the crib sheet. Sounds good, but here's the problem.

In October, with my mind on business and my customers, I forgot my crib sheet! Oh no, now what? Well, I had played some 9/6 Jacks on my home computer before I left, so I figured that I was up enough on the game to play a bit. So I did. And then it happened. I was dealt the following hand:

AD, JD, TD, 8D, TC

Well, I knew that the low pair was not the right play. I also knew that normally the 3 Royal was better than the 4-card Flush, but here I wasn't so sure. The AJT Royal is one of the weakest Royals one can have since it is a double inside straight. Worse, the discarded TC means trip tens are unlikely and the discarded 8D reduces the chance of a diamond Flush. Still, a Royal is tough to part with, especially in a progressive game. The Royal payoff was at $1080. Rats! Why did I forget that crib sheet?

Well, I kept the AJT Royal. I drew a 5D and QH, in that order. Hindsight is a wonderful thing; it appears I should have kept the 4 Flush. Not really. Hindsight can lead you astray and I know not to draw inferences from it. Nevertheless, I was bothered. Had I played the hand properly? Finally, I couldn't stand it any more, cashed out, went to my room and did the following calculation.

Keeping the AJT Royal gave me 1 chance for a Royal, 0 for a Straight Flush, 0 for quads, and 0 for a Full House. There were 9 diamonds remaining, so the number of diamond Flushes available was C(9,2) - 1 = 35, the - 1 for the Royal already counted. (Recall that C(n, k) = n!/k!(n-k)! and is the number of ways of choosing a set of k objects from a set of n objects.) There are 4 x 4 ways to fill the A-T straight, but again we have to subtract 1 for the already counted Royal. The two pair can occur as either AJ, AT, or JT. There are 3 Aces and 3 Jacks left in the deck, so AJ can occur 3 x 3 = 9 ways. There are only 2 tens left, so AT and JT each occur 3 x 2 = 6 ways. Altogether, then, two pair can occur 21 ways. Trips are easy. Either three Aces (C(3,2) = 3 ways), three Jacks (3 ways) or three tens (one way); total 7. To get a high pair, for example, we can have Aces. Here there are three ways to pick a matching Ace and for each of these choices there are 39 cards left in the deck that won't match anything else; total 117. Same for a pair of Jacks, 117. A pair of Kings can occur C(4,2) = 6 ways and likewise for a pair of Queens. Altogether there are 246 High Pairs. The number of different hands that we can form picking 2 cards from the remaining 47 is C(47, 2) = 1081. I actually calculated the number of losers directly so that I could double check my calculations, but since I now know that the above are correct we can just add the number of winners and subtract this total from 1081 to get the number of losers; it is 756. All done. Wow, do I know how to have a wild time in Las Vegas or what?

Very well, here was my expected return on a per coin basis by keeping the AJT Royal:

Now suppose I had kept the AJT8 diamond Flush. Here we just draw one card of 47. The result cannot be a Royal, a Straight Flush, Quads, a Full House, a Straight, Trips, nor Two Pair. There are 9 Flushes and 6 High Pairs; that's it. Now our table looks like this:

By keeping the AJT Royal my expected return per coin played was 1443/1081 or 1.3349 coins. On the other hand, had I kept the AJT8 Flush my expected return per coin would have been 60/47 or 1.2766 coins. My decision was a correct but lucky guess. Suppose, however, that I had not been playing a progressive machine. In this case the Royal payout in my first table would have been 800 instead of 864. The total in the product column would have been 1379 so the coins returned per coin played would have been 1379/1081 or 1.2756; the correct play would have been to hold the AJT8 Flush. Did the professor screw up? You bet I did; I took a guess instead of knowing.

Video Poker is a tough game! Notice that playing a progressive game is different from playing a fixed schedule game. So, pick your game and stick to it. If you make up a crib sheet, it must be tailored to the specific game you play. The above type hand was correctly listed on my crib sheet, but it sure didn't do me much good sitting back home. See you next month.

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