Probability of 4 to a straight or a completed straight on the board

[bilko]

Hi

Can someone tell me what the probability of this is after the river card in holdem?

e.g A23x5, 89TJx, 45678, 9xJQK


Thanks

Bilko

[bigpooch]

There are C(52,5) = 2598960 combinations of board cards that
are possible. Some might recall these common numbers from
draw poker:

40 straight flushes
10200 straights


Now it remains to calculate four-card straights. There are
basically two types, four in a sequence and those with one
gap inside (gutshot). Of the first type, the sequences with
an ace do allow an extra rank which won't produce a five
card straight on board. There are 11 sequences and 10x3=30
one-gappers. The following include 5 card flushes on
board; if you want to exclude those, take away 4 of the 1024
suit combinations allowed for the nonpaired boards below.

No pair:
2x(8x1024) + 9x(7x1024) + 30x(8x1024) = 319 x 1024 = 326656
(five card flushes: 319 x 4 = 1276 so those without a five
card flush on board 325380)

Pair:
(11+30)x4x6x64 = 62976

Summing, 10200+325380+62976 = 398556.

Altogether, excluding five-card flushes on board, there
are then 398556 possibilities yielding a probability of
about 0.1533521. This explains why when playing draw poker
everyone seems to notice getting a lot of straight draws,
especially gutshots!

Now, if you include five-card flushes (don't know exactly
why you would), there would be an additonal 40+1276 = 1316
hands making the total now 399872 hands and the probability
0.1538585.

[bilko]