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# Three Card Poker Rankings

6 March 2005

This month I want to use the material we developed in my January article to explain how one arrives at the rankings in the game of Three Card Poker. Although these rankings have frequently been listed by me and others, you now have the mathematical tools to explain just how one arrives at them. Before we get to that, however, I want to mention an upcoming event that I think you'll want to consider.

This Spring, May 21st and 22nd, Frank Scoblete is holding his Gambler's Jamboree at the Casino Windsor in Windsor, Ontario, Canada (minutes from Detroit). There will be over 42 seminars and hands-on individual instruction at this event. I am giving two of the seminars: one on "How casinos get the edge on you and what you can do about it" and the other on "Casino games versus Lottery games." I assume that since you are reading this column you have an interest in the mathematics of gambling. Well, I'll be there and happy to meet with you in person to discuss any topic that interests you.

Hope to see you there!

Now about those Poker hands. In Three Card Poker there are no Quads, Full Houses, or Two Pair hands. So, the hands on which I want to concentrate are Straights, Flushes, and Triples.

For five card Poker hands there are 40 Straight Flushes (including the Royal). This is easily seen by noting that the lowest card in the Straight Flush can be the Ace through Ten. Once the suit is chosen (four ways) and the low card (ten ways), there is only one Straight Flush possible. Thus, there are 40 of them. As mentioned in my January article there are 2,598,960 hands in all so getting a straight flush is not too likely.

To calculate the number of five card flushes it is convenient to think of our 52 card deck as being comprised of 4 packets of 13 suited cards each. We pick a suit (4 ways) and then pick 5 of the 13 in that suit, C(13,5) ways (check my December article for an explanation of this notation). Now

C(13,5) = (13 x 12 x 11 x 10 x 9) / (5 x 4 x 3 x 2 x 1) (1)

or

# Flushes = 4 x C(13,5) = 4 x 1287 = 5148 (2)

Forty of these are Straight Flushes so the number of plain Flushes is 5108.

To calculate the number of 5 card Straights we note that there are 10 possible low cards, Ace through Ten. Thinking of our 52 card deck as 13 packets of 4 equally ranked cards each we choose the low rank (10 ways) and then for each of the cards in succession we choose one card (four ways each) so altogether

# Straights = 10 x 45 = 10,240 (3)

Once again, 40 of these Straights are Straight Flushes and have already been counted so there are 10,200 plain Straights.

To calculate the number of 5 card Triples it is convenient to think of our 52 card deck as consisting of 13 packets as we did in the Straight calculation. Here we pick one of the 13 and choose 3 cards from it; there are 13 x C(4,3) or 52 ways to do this. Of the remaining 48 cards we want to pick two (so we don't get Quads). But wait, we don't want to pick a pair either or else we would have a Full House. The solution is to pick two of the remaining 12 packets (C(12,2) or 66 ways) and then pick one card from each (C(4,1) ways each). Altogether, then

# Triples = 52 x 66 X 4 x 4 = 54,912 (4)

Let us summarize our results:

 Hand Number Flush 5108 Straight 10,200 Triples 54,912

Certain Five Card Poker Frequencies

These results are consistent with the usual Poker rankings.

Now let's look at Three Card Poker. First of all there are C(52,3) or 22,100 Three Card Poker hands. The Straight Flushes (including Royals) can consist of 12 low cards (Ace through Queen) and 4 suits so there are 48 of them. For the triples there are 13 choices for the card rank and then C(4,3) ways of choosing three of that rank. C(4,3) = 4 so there are 4 x 13 or 52 Triple hands.

For the straights there are (as in the Straight Flushes) 12 cards that can be low in the hand, Once that rank is chosen there are four ways of choosing the lowest, four ways of choosing the middle, and four ways of choosing the highest card in the hand. The number of Straights, therefore, is

# Straights = 12 x 4 x 4 x 4 = 768 (5)

As I the five card case, 48 of these are Straight Flushes and have already been counted so there are 768 - 48 or 720 plain Straights.

Finally, for the Flushes, there are 4 ways to choose the suit and once this is done there are C(13,3) ways to choose 3 cards of the thirteen suited. Altogether then

# Flushes = 4 x C(13,3) = 4 X 286 = 1144 (6)

Again, 48 of these are Straight Flushes so there are 1144 - 48 or 1096 plain Flushes.

In summary we have

 Hand Number Triples 52 Straight 720 Flush 1,096

Certain Three Card Poker Frequencies

I guess you can see why Three Card Poker hands are ranked the way they are. It is interesting to note that a Triple is almost as difficult to obtain as a Straight Flush.

If you would like to know how to calculate frequencies for any of the other types of hands for either three or five card Poker you can write to me at 711cat@comcast.net.

I'll have another gambling problem using the combination formula C(n, k) in next month's article. In the meantime don't forget about that Gambler's Jamboree in May; it's bound to be an educational but fun filled weekend. 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