36 cards deck poker

Hi all,

My first post here, so first thing first, congrats to the wizard for great site, I love the odds :)!
I'm hoping to find some help here as I don't manage to calculate myself following probabilities....

The number and probabilities for holdem hands in a 52 cards decks are the following. (ty google:))

hand number Probability
straight flush 41,584 .00031
4-of-a-kind 224,848 .0017
full house 3,473,184 .026
flush 4,047,644 .030
straight 6,180,020 .046
3-of-a-kind 6,461,620 .048
two pairs 31,433,400 .235
pair 58,627,800 .438
high card 23,294,460 .174

Does someone know what they are with a 36 cards deck ?
I mean if:
- the deck is 36 cards 6,7,8,9,T,J,Q,K,A and 4 suits heart diamond, spade and clubs
- I take 7 cards of the deck
- out of these 7 what is the probability for each poker hands (5 cards)
- where here A,6,7,8,9 is the lowest straight possible and T,J,Q,K,A is the highest straight possible

Thank you for your help,
Pokerguy

Sounds like you are referring to Six Plus Hold' em, a game popular in Macau and elsewhere.

I can't tell you the exact probabilities, but in this game, a flush beats a full house, and three of a kind beats a straight.

A game in which A,6,7,8,9 is a straight should offend anyone's poker sensibilities.

Quote: MathExtremist

A game in which A,6,7,8,9 is a straight should offend anyone's poker sensibilities.

Got my attention. lol...

I'll do a couple now.

N(flush) = 4*[C(9,5)*C(27,2) + C(9,6)*27 + C(9,7)] = 46530
P(flush) = 46530 / C(36,7) = 47/8432 =~ 0.5574%

N(boat) = 9*8 * C(4,3)*C(4,2)*C(28,2) = 653184
P =~ 7.82%, about 14x as frequent as a flush which is why it's worth less.

Hi Almondbread,

Thx for your stats even if now I'm even more confused that before. :)
Below is what I have for full house(boat) and flush....you'll see my results are not quite same as yours. :)
I'm really not sure who is right or wrong...that's why I'm hoping to find some input here.

N(Flush) = (4*(C(9,7)-(4+7+10))+(4*(C(9,6)-(5+14))*27)+(4*(C(9,5)-6)*C(27,2))=175560
P(Flush) = 175560 / C(36,7)) = 175560 / 8347680 = 2.103%

N(Full house) = (C(9,2)*C(4,3)*C(4,3)*28)+(9*C(8,2)*C(4,3)*C(4,2)*C(4,2))+(9*8*C(7,2)*C(4,3)*C(4,2)*4*4)= 633024
P (Full house) = 633024 / (C(36,7) = 633024 / 8347680 = 7 .583%

Still explains why Full house is worth less :)

Hope someone can help here. :)

Sorry, both of my stats are wrong. For the flushes I forgot to subtract straight-flushes. For the boats I double-counted the cases of trips+twopair and I forgot to count the cases of double-trips.

N(straight flush) = 4*[C(31,2) + 5*C(30,2)] = 10560
P = 2/1581

N(flush) = 46530 - 10560 = 35970
P =~ 0.4309 %

N(boat) = 653184 - 3*C(9,3)*C(4,3)*C(4,2)^2 + C(9,2)*28*C(4,3)^2 = 633024

Since we still disagree on the flushes, I'll look at it again tomorrow.

Edit: In your flush calculation, where does 4+7+10 come from, and 5+14 and 6?

I calculate the following:

Best 5-Card Hand	Combinations	Probability
Royal Flush	1,860	0.000223
Straight Flush	8,700	0.001042
Four of a Kind	44,640	0.005348
Full House	633,024	0.075832
Flush	175,560	0.021031
Straight	1,169,940	0.140152
Three of a Kind	607,200	0.072739
Two Pair	3,157,056	0.378196
One Pair	2,316,600	0.277514
High Card	233,100	0.027924
Totals	8,347,680	1.000000

My analysis assumed that standard poker hand ranking applies. For example, 89TJQQQ is a Straight, not Three of a Kind.
Results were calculated by writing a program which iterated through all combin(36,7) hands and determined the best 5-card hand.

Oh, for my flushes I wrote 4*[...] but in my calculator I forgot the 4 so my result was off by a factor of about 4. Now that I typed it correctly (and subtracted straight flushes), I too get an answer of 175560 combos.

Thank you JB for that.

I'm having a hard time to accept that there are more combinations to make a full house than to make three of kind.
It's certainly due to what you're explaining about standard poker hand ranking applies where 89TJQQQ is a Straight and not Three of a Kind. It's really not intuitive at all. :)

Quote: pokerguy

I'm having a hard time to accept that there are more combinations to make a full house than to make three of kind.

Keep in mind that there are only 9 ranks, and you're using 7 cards to make a 5-card hand. When 3 of the 7 cards are the same rank, it is much more likely that the remaining 4 cards will contain a pair.

On a recent theme, this deck could be the basis for several funky bar bets. Even money that 7 cards will show 2 pair vs. 1 pair, etc.