help to calculate the best hands in 3 card Guts

[theRealMacoy]

Hello Gentelmen,

I have never posted here but I would love some help with calculating hand ranking when using only three cards.

I played in a home game tonight where we played 3-card Guts at the end. Not to mention a few twists to the play of the game they also included straights, flushes and straight flushes into the rankings of hands.

The winning hands were ranked as follows:
(from best to worst in descending order)

three of a kind
straight flush
flush
straight
pair
high card

I was wondering what the odds where to get these hands with three cards. Given that we saw two flushes and a straight flush in the 6 hands we played I was thinking that these rankings might not be correct.....although perhaps it was just randomness showing it's quirky self again.

Any insight to the wonderful world of probability would be great.

Cheers,
the Real Macoy

[BruceZ]

[ QUOTE ]
Hello Gentelmen,

I have never posted here but I would love some help with calculating hand ranking when using only three cards.

[/ QUOTE ]

Here are the numbers of ways to make each hand, along with the odds against each hand. If we add up all the numbers of hands, the sum equals the total number of hands 22,100 = C(52,3) as it should, so this serves as a check. To get odds, divide 22,100 by these numbers and subtract 1. Hands are listed in descending order. Three-of-a-kind beats both flushes and straights, and a straight beats a flush.


straight flush: 4*12 = 48; 459-to-1

three of a kind: 13*4 = 52; 424-to-1

straight: 12*(4*4*4) - (48 straight flushes) = 720; 29.7-to-1

flush: 4*C(13,3) - (48 straight flushes) = 1096; 19.2-to-1

pair: 13*6*48 = 3744; 4.9-to-1

high card: (52*48*44)/6 - (720 straights) - (1096 flushes) - (48 straight flushes) = 16,440; 0.34-to-1