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Four Card Poker Paradox

5 July 2003

I'm sure you've all heard that old admonition "Be careful what you say, your words may come back to haunt you." Well, my words did come back to haunt me.

Back in March of 2001 I wrote an article entitled Four Card Poker? I Don't Think So. In that article I argued that the frequencies of four-card Poker hands are such that I thought players would be uncomfortable with the natural hand rankings. In case you missed, the article here is a table showing the frequencies of four card Poker hands.

Hand Frequency
Royal Flush 4
Four of a Kind 13
Straight Flush 40
Three of a Kind 2,495
Straight 2,772
Two Pair 2,808
Flush 2,816
Pair 82,368
High Card 177,408
Total - 270,725
Figure 1
Four Card Poker Frequencies

You see that the usual Poker hierarchy is drastically changed when four-card hands are used. Three of a Kind beats a Straight and Two Pair beats a Flush. What is more, the frequencies for Flush through Three of a Kind are very close. It was my contention that Poker players would be uncomfortable with these rankings. I still believe that. Nevertheless, in the past couple of months two different four-card Poker games have come to my attention, one of which, I understand, has been playing for over a year now. So do I have to eat crow on this?

Well, I came close to eating crow but fortunately I did mention in my article that one could produce different numbers by having the player choose the best four-card hand from 5, 6, or 7 cards (although I tempered the remark by indicating that a player would not be happy turning a Full House into Two Pair). So, crow is not on my bill of fare since both of the aforementioned games deal five-card hands and have the player (or dealer) select the best four-card Poker hand from the five. Let's see what this does to the frequencies.

How many Straight Flushes (including Royals) are there? Well there are four suits and the straights in each suit occur as A-J down to 4-A so there are eleven of them. The A-J can be paired with any of the 48 remaining cards; the other 10 can be paired with only 47 of the remaining 48 since, for example, putting a suited J with a 10-7 Straight Flush would produce a J-8 Straight Flush rather than the desired 10-7. Hence there are 4 x 48 + 4 x 10 x 47 or 2072 Straight Flushes. There are 13 four-card Four of a Kinds and any one of the remaining 48 cards can be paired with each to make a five-card hand, so there are 13 x 48 or 624 of these. There are 13 choices of ranks for a Three of a Kind and for each such choice there are four ways to pick the three from the four. Picking 2 of the remaining 48 (1128 ways) we have the number of five-card hands containing a Three of a Kind is 13 x 4 x 1128 or 58,656. Now here is where things get interesting. The two games mentioned above rank hands as follows:

Game #1: Straight Flush
Four of a Kind
Flush
Straight
Three of a Kind
Two Pair
One Pair
High Card
Game #2: Four of a Kind
Straight Flush
Three of a Kind
Flush
Straight
Two Pair
One Pair
High Card

Clearly the inventor of Game #1 wanted to keep the hand rankings the same as they are in regular Poker even though, as we will see, the natural rankings by frequency are different. Game #2, on the other hand, has the top three hands in the correct order according to frequency. Both games rank the Flush above the Straight. Is this correct? Well, note first that if the Flush is ranked above the Straight and we are faced with a situation wherein our five-card hand contains hands of both types, we should opt to pick the Flush rather than the Straight. The calculation is a bit tricky so I'm going to skip it - write to me if you would like details. It turns out that with this strategy there are 116,688 five-card hands that contain four-card Flushes. Of these 2,072 are Straight Flushes so subtracting these we end up with 114,616 five-card hands that contain ordinary Flushes. This leaves 101,808 five-card hands that contain ordinary Straights. So both of the above rankings appear to be in the wrong order. Here are all of the frequencies with the Flush listed above the Straight:

Type Hand Frequency
Four of a Kind 624
Straight Flush 2,072
Three of a Kind 58,656
Flush 114,616
Straight 101,808
Two Pair 123,552
One Pair 1,047,552
High Card 1,150,080
Total - 2,598,960
Figure 2
Hand Frequencies when Flush > Straight

Okay, so it looks like we have to rank the Straight higher than the Flush. This means that when faced with a five-card hand that contains both a four-card Flush and a four-card Straight that we should pick the straight rather than the Flush. The following table shows what happens when we do this:

Type Hand Frequency
Four of a Kind 624
Straight Flush 2072
Three of a Kind 58,656
Straight 110,464
Flush 105,960
Two Pair 123,552
One Pair 1,047,552
High Card 1,150,080
Total - 2,598,960
Figure 3
Hand Frequencies when Straight > Flush

Oh no! Now there are fewer Flushes than Straights. So there you have it. When four-card Poker hands are selected from fiver- card hands, it is impossible to rank the Straights and Flushes in an order than reflects their natural frequencies. Fascinating!

As a practical matter there is no harm in this paradoxical situation, however, I think that this same phenomenon will occur in other games wherein a hand is selected as being the best hand chosen from a larger hand. There may be some surprises laying in wait there and gaming developers should keep this in mind. See you next month.

Four Card Poker Paradox is republished from Online.CasinoCity.com.
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