Trying to Compute Nine-Card Probabilities
Nevermind, you already know about Durango Bill's
Quote: NowTheSerpentI've verified all the combinatiion totals at the Wizard's Poker Probabilities page up through Eight-Card Stud. I worked through Nine-Card Straight Flushes with no trouble, but I'm still unclear about the proper way to subtract SF's from Four-of-a-Kind and Full-House hands. For instance, naive computation using
13C3*3C1*4C1 = 3,432 [for all AAAABBBBC 4oak forms]
+ 13C3*3C1*2C1*4C3*4C2 = 41,184 [for all AAAABBBCC 4oak forms]
+ 13C4*4C1*3C1*4C3*(4C1)^2 = 549,120 [for all AAAABBBCD forms]
+ 13C4*4C1*3C1*(4C2)^2*4C1 = 1,235,520 [for all AAAABBCCD forms]
+ 13C5*5C1*4C1*4C2*(4C1)^3 = 10,543,104 [for all AAAABBCDE forms including the 10 A-B-C-D-E consecutives]
+ 13C6*6C1*(4C1)^5 = 9,884,160 [for all AAAABCDEF forms including the 71 5- and 6-rank consecutives]
gives a total of 22,256,520, which according the Wiz and Durango Bill is 8,904 too many. Obviously, the last two terms produce the redundancies, but what is the right way to subtract these SF's out?
Step one: fix a (copy and paste?) error:
AAAABBCDE should be 9,884,160
AAAABCDEF should be 10,543,104
Step two:
See how may straight flushes fit the pattern AAAABBCDE.
There are 40 differant straight flushes (10 differant ranks for the highest card and 4 suits to pick from)
The 4 of a kind can use any of the 5 cards in the straight flush.
The pair can be in the remaing 4 cards in the straight flush and there are 3 differant suits left.
40 * 5 * 4 * 3 = 2400 straight flushes that fit the pattern AAAABBCDE
Step three:
Tell miplet to get some sleep. If he doesn't reply in the next few days, send him a reminder pm.
Find the number of 6 card straight flushes that fit the pattern AAAABCDEF
There are 36 differant 6 card straight flushes (9 differant ranks for the highest card and 4 suits to pick from)
The 4 of a kind can use any of the 6 cards in the straight flush.
36*6 = 216
Step 5:
Find the number of 5 card straight flushes That fit the pattern AAAABCDEF Where BCDEF is the straight flush and AAAA does not make it a 6 card straight flush.
For Ace high straight flushes there are 7 ranks for the four of a kind that won't makea 6 card straight flush. Same for 5 high.
2*4*7= 56.
For the remaining 8 ranks, there are 6 ranks for the four of a kind that won't makea 6 card straight flush.
8*4*6 = 192
Step 6:
Find the number of 5 card straight flushes That fit the pattern AAAABCDEF Where AAAA is part of the straight flush and the singleton is not the same suit as the straight flush.
There are 40 differant straight flushes (10 differant ranks for the highest card and 4 suits to pick from)
The 4 of a kind can use any of the 5 cards in the straight flush.
The singleton can be any of the remaining 8 ranks in 3 remaining suits.
10*4*5*8*3 = 4800
Step 7:
Find the number of 5 card straight flushes That fit the pattern AAAABCDEF Where AAAA is part of the straight flush and the singleton is the same suit as the straight flush.
For Ace high straight flushes there are 7 ranks for the singleton that won't make a 6 card straight flush. Same for 5 high.
The 4 of a kind can use any of the 5 cards in the straight flush.
2*4*5*7= 280.
For the remaining 8 ranks for the high card in the straight flush, there are 6 ranks for the singleton that won't make a 6 card straight flush.
The 4 of a kind can use any of the 5 cards in the straight flush.
8*4*5*6 = 960
Step 8:
Add up the combinations:
2400 + 216 + 248 + 4800 + 280 + 960 = 8904
You are on your own for the rest. Just remember that doing these by hand is very error prone.
