Odds of low in 7 Card Stud 8 or Better

[therazz]

If we assume 6 players play every hand until being dealt the last card, what would be the odds of one or more players having 8 or better for low? Thank you.

There are 34,840,576 possible 7-card hands that contain a 5-card low that is an 8 or better.

There are 133,784,560 total 7-card hands.

(I can post these calculations if you like, but they are a bit tedious.)

Thus, the likelihood of a random hand to contain an eight-or-better low is 34,840,576/133,784,560 = 26.04%; and the likelihood that a random hand will not contain a low is 73.96%.

You might think that we have to worry about correlations between hands -- that if hand #1 gets all wheel cards, hand #2 is therefore less likely to catch low cards -- for figuring the answer to your question, but it turns out that we don't. In particular, the likelihood that at least one hand has a low is going to be 1 - (the likelihood that no hands have a low); and the likelihood that no hand has a low among six players will be 0.7396 raised to the 6th power. 0.7396^6 = 0.1637, and so the likelihood that one or more hand out of six will have a low is 1 - 0.1637 = 0.8363.

In short, there is an 83.6% chance that at least one of six hands will develop a low by seventh street, if none of them fold.

[AlanBostick]

[therazz]

Thank you very much Alan for your time and efforts, it is greatly appreciated.

[AlanBostick]

De nada. I took "the time and effort" the better part of a year ago. I just copied the answer out of my eight-or-better notes.

Alan,
Do you have any more little gems in your 7CS/8 notes?

Thanks,
KeithO