[BruceZ]

Dan,

I've computed all the exact answers accurate to the tenths decimal place (and in most cases better). This required going out to 3 terms in each case. As expected, they only changed by a few tenths from the approximate values given earlier. Since they are all in agreement with the approximate results, we can have confidence in this computation, and it would be better to use these more accurate numbers. If you want the the Excel spreadsheet I made which does this, PM me with your email. Here are the exact results:

1) KK vs. AA

9 handed: 1 in 25.6
4 handed: 1 in 68.1

2) QQ vs. AA,KK

9 handed: 1 in 13.0
4 handed: 1 in 34.2

3) JJ vs. AA,KK,QQ

9 handed: 1 in 8.9
4 handed: 1 in 22.9

4) AK vs. AA,KK

9 handed: 1 in 25.8
4 handed: 1 in 68.2

5) AQ vs. AA,KK,QQ

9 handed: 1 in 13.1
4 handed: 1 in 34.3


I made a minor correction to the formulas given in my first post for cases 1 and 4. In case 1 there are 6 ways 2 players can have AA, and in case 4 there are 18 ways 2 players can have AA or KK, not 9 ways. Here are the corrected formulas:

1) KK vs. AA

9 handed:

[ 6*8*P(48,14)/2^7 - 6*C(8,2)*P(46,12)/2^6 ] / [ P(50,16)/2^8 ] = 1 in 25.6.

4 handed:

[ 6*3*P(48,4)/2^2 - 6*C(3,2)*P(46,2)/2^1 ] / [ P(50,6)/2^3 ] = 1 in 68.1


4) AK vs. AA,KK

9 handed:

[ 6*8*P(48,14)/2^7 - 18*C(8,2)*P(46,12)/2^6) / [ P(50,16)/2^8 ] = 1 in 25.8

4 handed:

[ 6*3*P(48,4)/2^2 - 18*C(3,2)*P(46,2)/2^1 ] / [ P(50,6)/2^3 ] = 1 in 68.2

The other formulas are a little more complex, so I won't give them here, but they are on the spreadsheet.

-Bruce