Six Plus Hold'em is a variant of Texas Hold'em in which a short deck is used. A short deck is a deck in which all cards from 2 to 5 are removed from a standard 52-card deck, leaving only 36 cards. Due to the change in deck composition the frequency in which hands appear are also different, which is why hand rankings are different from standard poker. An official World Series of Poker event held a few months ago swapped Full House and Flush from their usual hierarchy, leaving the others untouched.
However a Wikipedia article about the variant claims the follwing about hand rankings:
Flush ranks higher than full house. In theory, three-of-a-kind ranks higher than a straight as the probability of achieving three-of-a-kind is lower than a straight in short-deck, however recent games have been ranking straight higher than three-of-a-kind which has become standard.
I was unconvinced that tournaments were arbitrarily defying mathematics so I decided to do the numbers myself.
Standard Deck
A. Flush
- Pick $5$ out of the $13$ ranks: $\binom{13}{5}=1287$. - Pick a suit: $4$. - Total: $1287\times 4=5148$. - This figure includes Straight Flushes.B. Straight
- Pick $1$ of $10$ rank combinations that constitute a Straight: A-5, 2-6, 3-7, ..., 10-A. - Assign a suit to each card: $4^{5}=1024$. - Total: $10\times 1024=10240$. - This figure includes $10\times 4=40$ Straight Flushes.C. Three of a Kind
1) Only a Three of a Kind and nothing higher - Pick one rank to be the Three of a Kind: $13$. - Pick two other suits: $\binom{12}{2}=66$. - Pick three out of four suits for the Three of a Kind: $\binom{4}{3}=4$. - Assign a suit to the other two cards: $4^{2}=16$. - Total: $13\times 66\times 4\times 16=54912$. 2) Full House - Pick the rank to be the Three of a Kind: $13$. - Pick another rank to be the Pair: $12$. - Pick three out of four suits for the Three of a Kind: $4$. - Pick two out of four suits for the Pair: $\binom{4}{2}=6$. - Total: $13\times 12\times 4\times 6=3744$. 3) Four of a Kind - Pick the rank to be the Four of a Kind: $13$. - Pick another rank to be the lone card: $12$. - Pick a suit for the lone card: $4$. -Total: $13\times 12\times 4=624$.Short Deck
A. Flush
- Pick $5$ out of $9$ ranks: $\binom{9}{5}=126$. - Pick a suit: $4$. - Total: $126\times 4=504$. - Includes Straight Flushes.B. Straight
- Pick $1$ of $6$ combinations: A-9, 6-10, 7-J, ..., 10-A. - Assign a suit to each card: $4^{5}=1024$. - Total: $6\times 1024=6144$. - This figure includes $6\times 4=24$ Straight Flushes.C. Three of a Kind
1) Only a Three of a Kind and nothing higher - Pick one rank to be the Three of a Kind: $9$. - Pick two other suits: $\binom{8}{2}=28$. - Pick three out of four suits for the Three of a Kind: $\binom{4}{3}=4$. - Assign a suit to the other two cards: $4^{2}=16$. - Total: $9\times 28\times 4\times 16=16128$. 2) Full House - Pick the rank to be the Three of a Kind: $9$. - Pick another rank to be the Pair: $8$. - Pick three out of four suits for the Three of a Kind: $4$. - Pick two out of four suits for the Pair: $\binom{4}{2}=6$. - Total: $9\times 8\times 4\times 6=1728$. 3) Four of a Kind - Pick the rank to be the Four of a Kind: $9$. - Pick another rank to be the lone card: $8$. - Pick a suit for the lone card: $4$. - Total: $9\times 8\times 4=288$.
To summarize:
| Full House | vs | Flush(Straight Flush) | |
|---|---|---|---|
| Standard Deck | 3744 | < | 5148(40) |
| Short Deck | 1728 | > | 504(24) |
| Straight(Straight Flush) | vs | Three of a Kind(Full House/Quads) | |
|---|---|---|---|
| Standard Deck | 10240(40) | < | 59280(3744/624) |
| Short Deck | 6144(24) | < | 18144(1728/288) |
As you can see the frequency of Full Houses compared to Flushes flips but Straights compared to Three of a Kinds does not, regardless of whether hands that also belong to a higher rank are deducted or not. Thus the hand ranking that WSOP used is mathematically correct. I read the reference of the Wikipedia article which claimed as follows:
In Six Plus straights appear in abundance, so much so, that they rank lower than three-of-a-kind (but still higher than two pair).
The article does not dive into the numbers so I have no idea about their reasoning. What method of measuring frequency could have led to this claim? Are there any errors in my logic or calculations?