Re: Odds question.
if you start with a pocket pair, then for both cards to play you must make quads in the rank you hold. this will happen C[48,3]/C[50,5] = 17296/2118760 = 0.816% of the time you start with a pocket pair.
If you start with any other hand, you will make quads where both cards play precisely when you get three of one of your ranks on the board and the other two cards below (or equal to, in most jackpots) the other rank in your hand.
for example, if you hold KJ, the odds of making quad J's with your K playing are C[43,2]/C[50,5] = 0.043%
this is derived by:
a) from the 3 remaining J's, choose all 3 (C[3,3] = 1)
b) from the 47 remaining cards, subtract out the ones that invalidate your kicker (the 4 A's) and choose 2 (C[43,2] = 903)
c) divide by number of possible boards (C[50,5]=2118760)
these are the chances of each rank playing as a kicker when the other card (x) makes quads, and x is the smaller rank:
Ax 0.05102%
Kx 0.04262%
Qx 0.03497%
Jx 0.02808%
Tx 0.02195%
9x 0.01657%
8x 0.01194%
7x 0.00807%
6x 0.00496%
5x 0.00260%
4x 0.00099%
3x 0.00014%
these are the chances of each rank playing as a kicker when the other card (x) makes quads, and x is the larger rank (if x is the larger rank, then four of the cards that would counterfeit your kicker are gone anyway!):
Kx 0.05102%
Qx 0.04262%
Jx 0.03497%
Tx 0.02808%
9x 0.02195%
8x 0.01657%
7x 0.01194%
6x 0.00807%
5x 0.00496%
4x 0.00260%
3x 0.00099%
2x 0.00014%
you can just sum up the numbers from these two columns to figure out any particular hand, for instance -- with AK you will make quads with a playing kicker 0.10204% of the time, with 27 you will make quads with a playing kicker 0.00821% of the time.
SO, if what you wanted to know was, if I play any two random cards, and see it to the river no matter what, without even looking, what percentage of the time will i flip it over at the end to find quads with a kicker that plays?
well, using the procedure above, excel tells me the answer is: 0.0942% of the time, or just shy of one in a thousand.
-switters
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