Distribution of Cards in Bridge

Date: 05/03/2001 at 19:27:47
From: David West
Subject: Bridge game - distribution of cards

In bridge, it often happens that you and your partner have 9 cards of 
a suit between you. In all the bridge books that I have read, it is 
stated that the probability of the remaining 4 cards splitting 2-2 is 
40.7%. However, when I try to prove this, I am lost. 

As I see it, there are 16 possible combinations for the 4 missing 
cards. I listed all the combinations, and see that in 6 out of the 16 
the cards would split 2-2 between the opponents' hands. 6 out of 16 
would be 37.5%, not the 40.7% quoted in all the bridge books. 

I am not ready to write off all the bridge books [and sites on the 
Web] that say 40.7%, so I am asking what the fallacy in my 6 out of 16 
combinations is.

At 74, I am really not a student any more, but your answer will help 
me get a good night's sleep. <g>

Thank you,

David West
rip@montrose.net

Date: 05/04/2001 at 14:07:54
From: Doctor Schwa
Subject: Re: Bridge game - distribution of cards

Hi David,

Your counting method is a good approximation (37.5 is close to 40.7),
but it does have a small flaw.

The flaw is that once you've put two or three cards of the suit in one
person's hand, there are fewer cards left in that hand, so the chances 
of that third or fourth card going there are lower. That is, more even 
splits are more likely; it's not just the same as tossing a coin. It's 
more like having a collection of 26 coins, 4 of which are heads, and 
taking 13 of them... in fact that's exactly what it's like!

Anyway, once you've gotten a few heads, it's less likely that the next
coin you pick will be heads, both because the pile is running short
of heads and because your hand is already getting full so there
are fewer coins left for you to pick.

To calculate the exact odds of the 2-2 split, I would take the number 
of hands that contain 2 of that suit (and 11 other cards) out of all 
possible 13 card hands.

That is, (see Permutations and Combinations from the Dr. Math FAQ, 
http://mathforum.org/dr.math/faq/faq.comb.perm.html   , if this doesn't 
make sense to you):

   (4 choose 2)*(22 choose 11) / (26 choose 13)

which on my calculator is 40.695652%

Similar calculations work for 3-1 and 4-0 splits.

Enjoy,

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/