I am doing research on the probability of improving various poker
hands.

If someone has rolled up trips on third street in a seven card stud
game, what are the chances of ending up with a full house or better.

I did 48*47*46*45 = 4,669,920 for the total number of permutations of
the last four cards. 47*44*41*38 = 3,221,944 for the total of
permutations that do not contain a full house or better.
3,221,944/4,669,920=.69 So there seems to be a 69% chance of not
improving.

My reasoning is faulty, though, I think because I'm supposed to be
using combinations, not permutations since order does not matter. I'm
not sure if I'm doing this right.

Please let me know the answer and the method you used to come up with
it.

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Request for Question Clarification by smudgy-ga on 31 Jul 2003 19:15 PDT

Hi jajogluck,

I think I might be able to answer this question, but I need a couple
of things clarified...

1: I am not up on my poker terminology: could you translate "trips on
third street"?

2: Can you provide a listing of what hands match or beat a full house
in seven-card stud?

Thanks,
smudgy.

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Clarification of Question by jajogluck-ga on 01 Aug 2003 13:00 PDT

trip is short for "triplet". rolled-up trips is three-of-a-kind in the
first three cards, a 1/425 shot.

By "full house or better" I am including the possibility of making
four of a kind (quads). Do not take the possiblity of catching a
straight flush into consideration, although that hand will beat quads,
since it's very very unlikely to happen. So lets ignore that
possiblity.

I think you muast be familiar with seven stud poker in order to
adequately answer this.

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Request for Question Clarification by justaskscott-ga on 01 Aug 2003 13:28 PDT

How many players are in the game?  I imagine that this would affect the odds.

You've got it nearly right.  After the third card dealt, there are 49
remaining cards in the deck.  The probability of NOT getting a full
house or 4-of-a-kind on the 4th card is 48/49 (only the one remaining
card matching those in your hand will do it.)  Assuming that card
doesn't match, there are four cards of the remaining 48 to give you
the boat or 4-of-a-kind on fifth street, or 44/48 cards that won't get
you there.  There are then seven cards (1+3+3) that would hit it on
sixth street, or 40/47 to fail.  Finally, if you've made it to the
last card and still need the boat or better, there are ten cards left
(1+3+3+3) out of the remaining 46, or 36/46 that it still won't
happen.

Multiplying the chances of it NOT happening, you get:
(48/49)*(44/48)*(40/47)*(36/46)=.598

Therefore the chances of you hitting the boat or better anytime along
the way is  .402, or roughly 40%.

Of course, the 4-of-a-kind case by itself is easy.  To get the four of
a kind, the one remaining card matching your trips needs to be within
the four cards to be revealed on the board, out of the 49 total
remaining cards.
4/49 = .0816, or 8.16% chance of pulling the four of a kind.

(Notice this is the same as calculating it similarly to the first
method, where the formula would be 1 - (48/49)*(47/48)*(46/47)*(45/46)
= 8.16%)

Hope this helps!

There is a great book on 7CS entitled "Seven Card Stud" by Konstantin
Othmer.  It breaks down the hand by street (3rd card, 4th card, etc.)
and the odds of making a flush, straight, full house, etc. by the end
of the game.
 
A great resource to practice poker is to play for free online.  Click
on or go to the following link for great FREE online poker software:
 
http://www.partypoker.com/index20100.htm?wm=2007906 
 
If you ever decide to play for real money online, use the sign-up
bonus code: "FULLHOUSE" and get 20% extra when you make your 1st
deposit.
 
Hope this helps...and good luck!