Two-deck poker

I was very bored today.

Poker traditionally has to be played with one deck. When you have more than one deck, the entire face of the game changes - more hands become possible, the probability of some hands changes drastically, and so on.

To demonstrate this, I decided to analyze poker with two decks of cards. As I calculate it, this is the correct ranking of hands:

Royal flush
Five of a kind
Straight flush
Flush with two pair
Four of a kind
Flush with one pair
Flush with no pairs (1)
Full house
Unflushed straight
Three of a kind
Two pair
One pair
High card

Note 1: You could, if you liked, join a flush with one pair and a flush with no pairs. Then, a full house would beat them. Flush with two pair is rare enough that it should stay separate, though.

All ties are handled in the same way they would be in normal poker.

Some notes:

• I tried to stick to the framework of basic poker hands as best I could. Because of this, five of a kind and flushes with pairs seemed like necessary additions. You could add all kinds of other spiffy hands if you wanted to (i.e. does having a suited pair make it better?), but that's beyond the point of my analysis.
• The Royal flush being top dog seemed like something players would expect, so I included it. If you preferred, you could consider the Royal to be a straight flush, with five of a kind being better. As long as the Royal flush is considered separately, however, it wins: there are 128 Royals and only 728 fives-of-a-kind.
• With five decks, the Royal flush gets dethroned as the best hand, because a flushed five of a kind (with only 52 possibilities) would be king. I'm not sure how adding more decks would affect this, however.

For the curious (and peer review), here are my actual calculations for each hand. "C" means combinations, e.g. 8C3 is the number of combinations for drawing 3 items out of a list of 8, derived from the formula nCr = n! / ( (n-r)! * r! )

Edit: I forgot to account for flushed vs. unflushed pairs. Embarrassing! The new numbers are correct.

Royal flush:
4 different suits to flush in
2 different ways of getting each card in the royal flush
2
2
2
2
128 different Royal flushes

Five of a kind:
13 different ranks
8C5=56 different ways of getting five cards of that rank
728 different fives-of-a-kind

Straight flush:
4 different suits to flush in
9 different high cards (since Ace high gives royal flush)
2 different ways of getting each card in that particular straight flush
2
2
2
2
1172 different straight flushes

Flush with two pair:
4 different suits to flush in
13C2=78 combinations of ranks for pairs
11 different ranks for fifth card
1 combination for first pair in suit
1 combination for second pair in suit
2 cards possible for fifth card in given rank
6884 different flushes with two pair

Four of a kind:
13 different ranks for four-of-a-kind
12 different ranks for fifth card
8C4=70 combinations for four-of-a-kind
8 different cards of rank for fifth card
87360 different fours-of-a-kind

Flush with one pair:
4 different suits to flush in
13 different ranks for pair
12C3=220 combinations of ranks for extra cards
1 combination for pair
2 ways of getting each extra card in chosen rank and suit
2
2
91520 different flushes with one pair

Flush with no pairs:
4 different suits to flush in
13C5=1287 combinations of ranks for five cards
2 ways of getting each card in chosen rank and suit
2
2
2
2
164738 different flushes with no pairs, including straight flushes
Subtract 1172 straight flushes
163566 different flushes with no pairs or straight

Full house:
13 different ranks for three-of-a-kind
12 remaining ranks for pair
8C3=56 combinations for three-of-a-kind
8C2=28 combinations for pair
244608 different full houses

Unflushed straight:
10 different high cards for a straight
8 ways of getting each card in the straight
8
8
8
8
327680 different straights, including straight flushes
Subtract 1172 straight flushes
326508 different unflushed straights

Three of a kind:
13 different ranks for three-of-a-kind
12C2=66 combinations for ranks of extra cards
8C3=56 combinations for three-of-a-kind
8 different cards for fourth card
8 different cards for fifth card
3075072 different threes-of-a-kind

Unflushed two pair:
13C2=78 combinations of ranks for pairs
11 different ranks for fifth card
8C2=28 combinations for first pair
8C2=28 combinations for second pair
8 different cards of rank for remaining card
5381376 different hands with two pair
Subtract 6884 different flushes with two pair
5374492 different unflushed hands with two pair

Unflushed pair:
13 different ranks for pair
12C3=220 combinations of ranks for extra cards
8C2=28 combinations for pair
8 different cards for each remaining extra card
8
8
41000960 different hands with one pair
Subtract 91520 different flushes with one pair
40909440 different unflushed hands with one pair

High card:
13C5=1287 combinations of ranks with no matches
8 ways of getting each card
8
8
8
8
42172416 different hands that do not contain a pair, three-of-a-kind, four-of-a-kind, or five-of-a-kind
Subtract 327680 different straights (including straight flushes)
Subtract 163566 different flushes with no pairs (not including straight flushes)
41681170 hands that contain no other hand

Quote: curtmack

I was very bored today.

I'll say! This reminds me of some of the stuff that floats around the internet. It's entertaining, but always leaves me thinking "Somebody has a lot of free time."

Quote: curtmack

i.e. does having a suited pair make it better?

Yes.

It's no different than the suited five of a kind in five deck poker that you mentioned.

Similarly, many Black Jack side bets pay X for specific cards, but pay more if they are suited.

Quote: DJTeddyBear

Quote: curtmack

i.e. does having a suited pair make it better?

Yes.

It's no different than the suited five of a kind in five deck poker that you mentioned.

Similarly, many Black Jack side bets pay X for specific cards, but pay more if they are suited.

Well, the flushed five of a kind is a special case: it's a flush, and it's five of a kind. Same with a straight flush (or for that matter, a Royal) in normal poker. You certainly could say that suited pairs are better, but keep in mind that there's a difference between, say, a suited pair of aces, and a flush with a pair of aces.

I didn't check all of your figures, but I disagree with your Two Pair / Three of a Kind result:


Three of a Kind
trips ..... combin(13,1)*combin(8,3) = 728
kickers ... combin(12,2)*combin(8,1)*combin(8,1) = 4224
total ..... 728 * 4224 = 3,075,072 (this agrees with your total)

Two Pair
pairs .... combin(13,2)*combin(8,2)*combin(8,2) = 61152
kicker ... combin(11,1)*combin(8,1) = 88
total .... 61152 * 88 = 5,381,376 (this is much higher than your total)


The above Two Pair figure does not subtract the counts for suited Two Pair hands if they are deemed to be higher in rank than other Two Pair hands. Nevertheless, the figures show that Three of a Kind is still a better-ranking hand than Two Pair.

Quote: JB

I didn't check all of your figures, but I disagree with your Two Pair / Three of a Kind result:


Three of a Kind
trips ..... combin(13,1)*combin(8,3) = 728
kickers ... combin(12,2)*combin(8,1)*combin(8,1) = 4224
total ..... 728 * 4224 = 3,075,072 (this agrees with your total)

Two Pair
pairs .... combin(13,2)*combin(8,2)*combin(8,2) = 61152
kicker ... combin(11,1)*combin(8,1) = 88
total .... 61152 * 88 = 5,381,376 (this is much higher than your total)


The above Two Pair figure does not subtract the counts for suited Two Pair hands if they are deemed to be higher in rank than other Two Pair hands. Nevertheless, the figures show that Three of a Kind is still a better-ranking hand than Two Pair.

Yeah, that looks right. I'm not sure where my mistake was, but it seems to be in punching numbers into my calculator. How I made the exact same mistake more than once is a bit weird, but whatever. I'll change it.

Quote: curtmack

Yeah, that looks right. I'm not sure where my mistake was, but it seems to be in punching numbers into my calculator. How I made the exact same mistake more than once is a bit weird, but whatever. I'll change it.

I think you missed the last factor of 8, which corresponds to the suit of the kicker. You listed it, but forgot to include it in the calculation.

I have a table for poker combinations with 1 to 8 decks here. Scroll down to "Multi-Deck Probabilities."

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The Casino I work at is getting a 6 deck Texas Hold'Em table game next month. 5 of a kind suited is the best hand. It's called Texas Shootout. It's reviewed on the Wizard of Odds site and looks pretty cool.

Quote: Zcore13

The Casino I work at is getting a 6 deck Texas Hold'Em table game next month. 5 of a kind suited is the best hand. It's called Texas Shootout. It's reviewed on the Wizard of Odds site and looks pretty cool.

Here's the Wiz's page on it: https://wizardofodds.com/texasshootout

Looks kinda interesting.

Where do you work?


On a side note: Would that be advertising? Nah. I'd bet that the Wiz wouldn't want you to mention your casino in every post, but since this is tied to the thread's topic, I doubt he'd mind a quick plug.

Yeah, that's why I didn't mention the name of the Casino in my original post. I'll asnwer your questions and if the Wizard feels it necessary to remove this post, I understand. I work at a small Casino in Prescott, AZ called Bucky's Casino. We're getting the game in about 2 weeks. I'll review it in the Table Games forum once things get rolling.

Feel free to name specific casinos whenever you want. In fact, I encourage it.

I found this post after someone asked about this in Reddit.

Spent the last few hours working through some of this, it's really interesting!

I think you've made a mistake though in the number of straight flushes; it should be 1152 not 1172. This then makes a couple of the other ones wrong.

Great job working through it though, it's a cool way to spend some time