Probability: Five and Seven Card Stud

Date: 6/7/96 at 10:53:44
From: Anonymous
Subject: Five card stud and seven card stud

We are students at a high school, and we are trying to figure out how 
to calculate the probability of getting two pair and three of a kind 
dealing with five and seven card stud.  We know the answers, but we 
would like to know how to calculate them mathemetically.  Is there  a 
formula?  

Thanks a lot.

Brian Staas, Ron Caputo, Jim Newton, and Ryan Mcleod. 

Date: 6/7/96 at 17:15:48
From: Doctor Anthony
Subject: Re: Five card stud and seven card stud

I am not a card game expert, but I will describe what I think you 
mean, and if I am wrong, you can probably adapt the mathematics to the 
actual situation you have in mind.

(1) Two pairs, say, Jack, Jack, five, five, and another card

The number of possible combinations of 5 cards from 52 is given by 
(52 C 5) = 2598960, and this will be the denominator in the 
probability calculations.

Number of ways of getting two Jacks from four is (4 C 2) = 6
Number of ways of getting two fives from four is (4 C 2) = 6
Number of ways of getting another card not a Jack or Five  = 44
Number of ways of selecting two face values from 13 = (13 C 2) = 78

Total number of ways of getting two pairs = 6*6*44*78 = 123552

Probability of two pairs = 123552/2598960

                         = 0.047539  

(2) Three of a kind, say Queen, Queen, Queen plus two other cards 
which must not be another Queen, or two of the same kind

Number of ways of selecting three queens from four = (4 C 3) = 4
Number of ways of selecting face value for three of a kind = 13
Number of ways of selecting face values for the other two cards = (12C2) 
   = 66
Number of ways of selecting the suits for the last two cards = 4^2 
   = 16  

Total number of ways of getting three of a kind = 4*13*66*16 = 54912.

Probability of three of a kind = 54912/2598960

                                 = 0.021128
                                 
-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/