Poker With a Pinochle Deck

Date: 7/1/96 at 9:29:41
From: Anonymous
Subject: Some what confused

Imagine that we are playing poker with a pinochle deck of 48 cards: 
two each of 9,10, J, Q, K, A in each of the four suits. If we are 
dealt a 5 card poker hand, what is the probability that it will be a 
full house?

Can you explain, please?  Thank you.

Date: 7/1/96 at 19:17:43
From: Doctor Anthony
Subject: Re: Some what confused

Full house = (for example) 2 aces + 3 jacks.

We could pick the face value of the pair of cards (aces in this case) 
in 6 ways, and then the face value of the threesome (jacks in this 
case) in 5 ways.  So face values could be chosen in 6 * 5 = 30 ways.

Now we must choose 2 aces from 8 and 3 jacks from 8 while there are a 
total of 48 cards from which we must select 5. So the probability of 
2 aces and 3 jacks is  

   (8_C_2)*(8_C_3)/(48_C_5).

This result must be multiplied by 30 to cover other possibilities for 
the face values of the cards making up the full house.

Required probability = 30*28*56/1712304
                     =  0.02747.

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