Double paired boards...

I was recently sitting at a $300 buy in no limit game and an interesting question arose. At first the answer seemed obvious, but after thinking about it for a while the answer began to appear hazy... The debate was whether it was more likely for a player at a table to have a full house on a single or double paired board. For instance: is it more likely that a player will have a Full if the board reads KK234 or KK223... Right off the bat I jumped in and said that it is far more likely that a player will have a full house on a double paired board. But on a single paired board, it seems that there may be more possiblilities to make a full house. If the board reads Kd Kh 2d 2h 3h a full can be made with any hand containing the one of the two black kings or one of the two black twos or the 3d 3s or the 3s 3c or 3d 3c.(I believe that my mistake is made here... should i count every possible hand combination which has a Black king or duece?) 7 possible hand combinations to make a full. With a board of Kh Kd 2h 3h 4h a full house can be made with a 4d 4s or 4s 4c or 4d 4c or 3d 3s or 3s 3c or 3d 3c or 2d 2s or 2s 2c or 2c 2d or Ks 2s or Ks 2d or Ks 2c or Ks 3s or Ks 3d or Ks 3c or Ks with all three combinations of 4's... My point being that although I would feel more uneasy about betting a flush on a double paired board than i would a single paired board i believe that there may be more possiblilities for a flush to be made on a single paired board, and if this is true than why would there not be more full houses on single paired boards than on double paired boards. My mind has been churning away at this one for a few days, and I am putting it into the hands (and heads) of the public now... I am done... <font color="blue"> </font>

[sthief09]

3322x

full houses are xx, 3y, and 2y. there are 3 ways of making xx, 80+1+3+4/2 = 86 ways to make each of deuces full or treys full (I split the 4 combos of deuce-trey for each), so there's 175 ways of making a boat there.

4432A

AA, 44, 33, 22 = 10
43, 42, A4 = 18

so 175 vs 28 or about 7 times more likely on a paired board