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Best of Donald Catlin

Gaming Guru

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Four Card Poker? I Don't Think So

4 March 2001

The 2000 World Gaming Congress and Expo was held in Las Vegas this past October and was bigger than ever. New games seem to be coming out of the woodwork. Continuing the trend of themed slot machines, such as Three Stooges, Elvis, Roller Derby, Yahtzee, Wheel of Fortune and the like, are machines based on TV's Jeopardy, Betty Boop, the old game of Battleship, Blondie, and so on. Table games with twists on Blackjack, Baccarat, Craps, and Poker were cropping up with a vengeance. In my own analysis business, I just finished two Poker-style games late last year and am now working on two new Poker-style games (all from four different developers). Some of these games deal five cards, some six, some seven, some have table cards, some have qualifying hands, etc., etc. There is, of course, a very successful game called Three Card Poker™ that is in play in many casinos around the country. It occurred to me that somewhere in this country someone will get the bright idea: "Hey, why not a game based on four-card Poker hands?" I'd like to explain why I don't think this will ever happen (God forbid if it already has) and in doing so illustrate why it is a good idea for gaming developers to have their game analyzed before dumping a lot of time and money into development.

To begin with, I'd like to turn back the clock and look at the development of what turned out to be a very successful game, namely Three Card Poker™. This game was the brain child of Derek Webb of Prime Table Games LLC™ in Las Vegas; I've mentioned Derek before in this column. The game has since been sold to Shuffle Master™ and continues to do well. This was a reasonable venture because a form of three card Poker had been around for years; I remember playing it long ago at someone's kitchen table. I believe it was called 'Spit' or some equally unsavory name; I really don't recall with certainty. What I do recall is that in this game a Straight beat a Flush. Let me show you why. There are C(52, 3) three card hands that one can deal from a 52 card deck [recall that C(n, k) = n! / k!(n - k)! and is the number of ways of selecting a set of k objects from a set of n objects; we have used this fact several times in these articles]. This number is:

C(52, 3) = (52 x 51 x 50) / (3 x 2 x 1) = 22,100 (1)

Now, I'm not going to calculate the numbers of all of the Poker hands here since we have done that type of calculation before. Let me, however, do a couple. First there are 4 Royal Flushes, that is easy, and 44 Straight Flushes (A23 through JQK), almost as easy. There are altogether C(4, 1) ways to choose a suit and then C(13, 3) to choose any three cards from that suit. The total number of 3 Flushes is thus

C(4, 1) x C(13, 3) = 4 x 286 = 1144 (2)

The number in (2), however, includes 44 Straight Flushes and 4 Royal Flushes, so we have to subtract these. The number of plain old 3 Flushes is 1096. For the straights, there are 12 card values that can be the low card in the straight (A through Q) and once the rank of the low card is chosen, there are 4 x 4 x 4 = 64 to build the straight. Altogether then, there are

# Straights = 12 x 64 = 768 (3)

Again, the 48 SFs and Royals are included, so there are really only 720 plain, old 3 Straights. Well, you get the idea.

Here is the way the hands break down in Three Card Poker:

Hand Number
Royal Flush 4
Straight Flush 44
Three of a Kind 52
Straight 720
Flush 1096
Pair 3,744
High Card 16,440
Total - 22,100
Figure 1
Three Card Poker Frequencies

Well, now you can see why the Straight beats the Flush. Notice that the margin is considerable, so it is a very clear decision. Also it is the only reversal from ordinary 5 card Poker (the Three of a Kind here rather naturally plays the role of the Quads in 5 card) so the trauma to one's Poker sensibilities is limited to this one rather compelling situation. Besides, as I mentioned above, the game with this feature has been around a while.

Now let me show you the unusual breakdown of 4 card Poker hands. There are C(52, 4) or 270,725 of them. I'll skip any calculations and just report the results; write to me if you want details.

Hand Number
Royal Flush 4
Four of a Kind 13
Straight Flush 40
Three of a Kind 2,496
Straight 2,772
Two Pair 2,808
Flush 2,816
Pair 82,368
High Card 177,408
Total - 270,725
Figure 2
Four Card Poker Frequencies

Look at that! Four of a Kind beats a Straight Flush, Three of a Kind beats a Straight, and worst of all, Two Pair beats a Flush. I don't think players would ever buy into these. Also disturbing is how close the numbers are for the Straight, Two Pair, and Flush. I think these results sink any possibility of creating such a game. Oh yeah, one could massage these numbers by adding a feature wherein the player had to chose the best four card hand out of 5, 6, or 7 cards. But can't you just feel the joy in turning a nice 5-card Full House into a four-card Two Pair? I don't think so.

My experience is that, when it comes to gambling games, the public doesn't like radical changes; they like things with which they are familiar. Now, maybe I'm wrong. Perhaps some enterprising developer will develop a successful four-card Poker game. I can, however, tell you who's not going to be that developer and who's not going to be doing the analysis. See you next month.

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