guts hand probabilities

[chocaholic]

I play in a home game where guts is played frequently, but with slight modifications. Each player is dealt 3 cards face down and 1 card face up, and the lowest card face down is wild. Hand rankings are quads, then trips, then a pair, no straights or flushes. I would like to know the odds of each hand happening, but not just trips in general; the odds of 3 aces, 3 kings, 3 queens, etc. Also, if possible, I would like to know how to get the answer; I've puzzled over it for a while with no success [img]/images/graemlins/smile.gif[/img] Thanks.

Dan

[filsteal]

Does that mean that each player's lowest face-down card is wild, or only that the rank of the lowest face-down card among all players' face-down cards is wild?

That is, if Abe's lowest hole card is a 2, Bob's is a 4, and Chuck's is a 5, are all three of those cards wild, or is only Abe's 2 wild?

[chocaholic]

sorry, I wasn't clear about that. All three of those cards are wild, but only for the player holding that card. If Abe's hand is, for example, 259 face down, then only the 2 is wild even though Chuck has a 5 as his lowest concealed card. Does that make sense?

[jesse8888]

The way I understand this, by this system each player has at least 1 wild card, correct? In the example, 2s are wild for Abe and 5s are wild for Chuck. Do I have that right?

[dave6]

Let's take 3 tens as an example.

First I'll introduce some notation: Capital X is a card higher than ten, lowercase x is a card lower than ten. The first card listed is face up. So, for example, T/TXx would mean you have a ten face up, and your face down cards are a ten, a lower card, and a higher card.

Now, just add up the different ways of making 3 tens:

x/TTT -- 32*4 = 128
X/TTT -- 16*4 = 64
x/xTT -- 32*31*6 = 5952
X/xTT -- 32*16*6 = 3072
T/XTT -- 4*16*3 = 192
T/xXT -- 4*32*16*3 = 6144
T/xxT -- 4*C(32,2)*3 = 5952

Total: 21504

The total number of possible combinations is 52*C(51,3) which is 1,082,900.

Probability(3 tens) = 21504/1082900 = 1.986%

The calculation is similar for other ranks, but you need to consider the differing numbers of overcards vs. undercards. Trip 2's is obviously the hardest to make, since there aren't any lower cards to use as a wild.