Probabilities of Hands in a Poker-like game with a 48 card deck with 6 suits of 8 cards each

Question

I'm working on designing a new card game. Instead of the traditional 52 cards of four suits consisting of 13 cards each, this deck will have 48 cards with six suits and 8 cards per suit.

I'm interested in implementing a poker-like game with this deck, similar to Texas Hold'em. For the rules, I want the order of the winning hands to be based on probability, so that the hardest to hit hands are ranked higher.

If you're familiar with Texas Hold'em, let's assume it's the same basic rules. Each player is dealt two face-down hole cards. Then the flop comes with 3 community cards, then a turn card is dealt, then a river. So there are 5 total community cards, plus 2 hole cards. Each player uses their hole cards plus the community cards to build the best possible hand.

This game will have an Ace that can function as both a 1 or a 9.

The winning hands in Poker, in order are:

Royal flush

Straight flush

Four of a kind

Full House

Flush

Straight

Three of a Kind

Two Pair

High Card

In our theoretical game, we could also have 5 of a kind. I'm not sure where that would fall in the rankings.

This is all possible to compute using math, right? Not requiring a computer simulation to predict?

Any insights would be greatly appreciated. Thanks!

Answer 1

I followed the advice of a commenter and looked at the Wikipedia page for how to calculate the best five card hands for ordinary poker, and then adapted the formulas as best I could to this theoretical game.

If you have a moment, please verify that my formulas are correct.

I'm going to use Excel notation because that's the "calculator" I used to solve these.

Total # of possible hands in the theoretical 48 card deck of 6 suits:

=COMBIN(48,5)

Which equals: 1,712,304

Straight Flush

=COMBIN(5,1) * COMBIN(6,1)

Which equals: 30

Royal Straight Flush: 6

Five of a Kind

=COMBIN(8,1) * COMBIN(6,6)

Which equals: 8

Four of a Kind

=COMBIN(8,1) * COMBIN(6,6) * COMBIN(5, 1) * COMBIN(6, 1)

Which equals: 240

Full House

=COMBIN(8,1) * COMBIN(6,5) * COMBIN(7,1) * COMBIN( 6,2)

Which equals: 5,040

Flush

=COMBIN(8,5) * COMBIN(6,1) - 30

Which equals: 306

Straight

=COMBIN(5,1) * ( COMBIN(6,1) ^ 5 ) - 30

Which equals: 38,850

Three of a Kind

=COMBIN(8, 1) * COMBIN(6,5) * COMBIN(7, 2) * ( COMBIN(6,1) ^ 2)

Which equals: 38,288

Two Pair

=COMBIN(8,2) * ( COMBIN(6,2) ^ 2 ) * COMBIN(6,1) * COMBIN(6,1)

Which equals: 226,800

Pair

=COMBIN(8, 1) * COMBIN(6,2) * COMBIN(7, 3) * ( COMBIN(6,1) ^ 3 )

Which equals: 907,200

So in conclusion the correct hand ranking order should be: Royal Straight Flush, Five of a Kind, Straight Flush, Flush, Full House, Three of a Kind, Straight, Two Pair, Pair and High Card.

Answer 2

Too long for a comment on your answer.

First: congratulations on working these out for yourself.

I haven't checked all the calculations. I think you have to think things through a little more, not just mimic the wikipedia calculations. (You are probably right enough to get the order of the hands correct.)

For example, for four of a kind I get

COMBIN(8,1) * COMBIN(6,4) * 7 * 6

which is $5040$.

The first factor chooses the kind, the second chooses the four suits. That leaves one more card, which can't be one of the kind, so $7$ choices in any of the $6$ suits. For clarity and consistency you could write the $7$ as COMBIN(7,1) and the $6$ as COMBIN(6,1).

I suggest you check the rest of your answers by writing that kind of paragraph for each calculation, as the wikipedia page does for ordinary poker.

Also: I think you should subtract the royal straight flushes from the count of straight flushes, so there are $30-6=24$ straight flushes.