6-card straight flushes

[cavalier]

I saw my first 6-card straight flush last night ( 4 through 9 ) in 7-stud 8-or-better and am wondering how (un)likely it is that I'll see another one.

The total number of 7-card poker hands is (52,7) = 133,784,560

When the smallest card in the 6-card straight flush is an ace, we can have 46 other cards as the 7th card. However, we are not interested in 7-card straight flushes, so we must eliminate the 7s. This leaves us 45 cards. This gives us 4*45 = 180 6-card straight flushes where the smallest card is an ace.

If the smallest card is 2 through 8 (7 cards * 4 suits = 28 cards) in our 6-card straight flush, then we may choose any 1 card other than the 2 cards that make it a 7-card straight flush. This gives us 28*(47-2) = 1260 straight flushes.

If the smallest card is a 9, we have the same case as if it were an ace since only 1 card must be ignored to make it a 7-card striaght flush. Therefore, there are 180 of these as well.

Total number of 6-card straight flushes is (180*2)+1260 = 1620

Probability of seeing one are : 1620 / 133,784,560 = .00001211 or 1 in 82583 hands

Given that the probability of seeing a Royal Flush is .0003232, we now know that we should expect to see about (0.00003232/0.00001211) = 2.67 Royal Flushes for every 6-card straight flush we see in 7-card stud.

p.s. If any of this math is wrong, please correct me. I am fairly new to this sort of figuring.

p.p.s. By the way, he caught the 7 on the river and he did scoop the pot with his 8-low. [img]/forums/images/icons/grin.gif[/img]

[BruceZ]

If the smallest card is 2 through 8 (7 cards * 4 suits = 28 cards) in our 6-card straight flush, then we may choose any 1 card other than the 2 cards that make it a 7-card straight flush. This gives us 28*(47-2) = 1260 straight flushes.

28*(46-2) = 1232

[cavalier]