Quote:
Originally Posted by JohnnyGroomsTD
These probabilities are for a 7 card hand.
I'm gonna argue that point, but with only 75% confidence (since again I wasn't a math major...)
You show the "count" of hands, along with the total at the end for choosing 5 cards from the 36-card deck, but not the count of the hands when choosing a full 7-cards and making the best 5-card hand out of those.
Let's look at the trips to see how this is much different:
Some portion of those 16,128 5-card trips hands are "counterfeited" by a card in slots 6 or 7 for full 7-card hands, making them into boats or quads or flushes or straights. This number is not insignificant, as is shown by a simple example.
Let's say we have 6
6
6
7
8
as a sample 5-card hand (rainbow.)
There are (31*30)/2 = 465 7-card hands that have this specific 5-card hand in it. (31 remaining cards, choose 2)
Of those, how many do not have a 6, 7, or 8 in any of the last 2 cards? I might be wrong here, but if we take all the remaining 6, 7, and 8 out of our 31-cards and choose 2, we have
(24*23)/2 combos left, or 276.
Additionally, we must remove the combinations that make straights, so all A9 and 9T variants. (4*4 each) And let's not forget the runner-runner pairs (6*6). So take 16, 16, and 36 away from 276 = 208
208 / 465 = 44.73% left!
Therefore, at most 45% of the times that 5 of our cards are 6
6
6
7
8
will we actually have a trips hand when 7 are pulled.
I posit that this invalidates the 5-card hand counts as a relevant basis for hand ranks in this game, as they are not necessarily accurate reflections of the 7-card world.
Some Googling found a link that I cannot verify, but it could show why this game is definitely a GAMBLING game when played with that rank table. The totals are correct at least (36 choose 7 = 8.3MM) as is the Royal (4 * 31 choose 2 = 1860) but I'm not smart enough to do the rest.
Link to thread
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
Flush 175,560 0.021031
High Card 233,100 0.027924
Three of a Kind 607,200 0.072739
Full House 633,024 0.075832
Straight 1,169,940 0.140152
One Pair 2,316,600 0.277514
Two Pair 3,157,056 0.378196
Totals 8,347,680 1.000000
~
Can you imagine hi-card beating trips!? Now I'm sure no casino is spreading the game this way... also, I have no idea what kind of dynamic is introduced by the fact that the board is shared - but I'm guessing it's not different than holdem.
Once again - someone better at math, or with a reliable source, I'm very interested in seeing a more complete proof or refuting of this conclusion! This seems pretty important if the MD gaming commission is going to rule on this; I have little faith they will understand it fully.
(Hey, they approved 6:5 BJ after not allowing it at first... so someone isn't doing the math.)