Odds of first 3 cards low in 7 Stud Hi-Lo

[therazz]

I really appreciate the time and effort by players responding to my questions.

What are the odds of being dealt the following low hands in your initial 3 cards: (1) 8-or-better (2) 7-or-better (3) 6-or-better (4) three cards to a wheel

Assuming an eight-handed game:
How many players will be dealt three cards 8 or lower?

[AlanBostick]

This is easy.

There are 32 cards (8 ranks of 4 each) that are 8 or lower; 28 that are 7 or lower; 24 that are 6 or lower; and 20 that are five or lower.

Let's look at three cards smaller than 8: There are 52 choose 3 = 52!/(3!*(52-3)!) = 52*51*50/6 = 22,100 three-card starting hands (I'm ignoring which one happens to be the upcard.)

A three-card low won't have a pair in it, so there are 32 first cards, 28 second cards, and 24 third cards that make up a 3-card low that is an 8 or better. The total number is going to be 32*28*24/6 = 3,584 possible hands (again, neglecting the order in which they are dealt). The probability of getting a low
start is 3584/22100 = 0.1622, or 16.2%. (The odds are 18516:3584 against, or 5.17:1 against.)

Likewise, there are 28*24*20/6 = 2240 starters that are 7 or better (P = 0.1014; 19860:2240 against, or 8.87:1 against); 24*20*16/6 = 1280 starters that are 6 or better (P= 0.0579; 20820:1280 against, or 16.27:1 against); and 20*16*12/3 = 640 starters consisting of unpaired wheel cards (P = 0.290; 21460:640 against, or 33.53:1 against.)

The exact expected number of low starters in an eight-handed deal may seem to be tough to compute, but in fact it turns out to be 8 * (probability of one hand being a low starter).

8 * .1622 = 1.2976.

In other words, most often just one player will have a low start, occasionally two, and three or more only infrequently.

This suggests an obvious measure of a good game: if multiway pots are common, your opponents are playing too many hands, and starting-hand selection starts you off with a big edge.

Is this helpful?

[therazz]

Once again, thank you Alan. Very helpful.