What are the odds of low in Stud 8 or Better

In a loose game of 7 Card Stud Hi-Lo with eight players being dealt in, what are the approx. odds that one or more hands will qualify with an 8 or better for low?

[Andy B]

Post stud/8 questions in the stud forum.

I don't think that it's possible to answer your question with any degree of confidence. It would depend on just how loose folks play, how willing they were to call on third with a bad card, on fourth with a bad card, etc. I think that there are too many variables at work.

What would be possible is to calculate the odds of someone making a low with seven players (not eight because you'd run out of cards) staying in until the river every hand. Or you could run a simulation with some very large number of hands or something.

[Buzz]

therazz - My (very limited) experience is that people tend to draw to lows in stud-8, even when they do not have very good low draws. Thus I think most starting hands that could end up qualifying for low tend not to be folded.

A couple of years ago in preparing for a tournament where stud-8 was one of the games being dealt, I wondered about the very question you’re posing and cranked out an answer just to give myself an idea of what to expect. As I recall, at the time I wanted to get some practice in a stud-8 ring game, but I couldn't find a stud-8 ring game. Instead, I read the first part of Ray’s Zee’s book and did the calculation about which you are asking. Something like that.

I’ll share my stumbling solution to the problem with you, but I can’t guarantee it’s correct.

If you play no-fold-’em-low stud-8, you should have a qualifying 8-or-better-low hand roughly once every 5.5 deals. (probability = 24390144/133784560 = 0.1823).

With seven players around the stud-8 table, all playing no-fold-’em-low, I figure someone should qualify for low roughly three hands out of four.

In a tournament, near, or at the final table, with competitors playing more tightly, it would not be unusual for no one to qualify for low most of the time. (That's my experience).

You asked about a table with eight players. I actually have played the game (stud-8) at a table with eight players on more than one occasion. Each time I vaguely wonder what will happen if everybody stays in, but as it has turned out, that hasn't happened yet. I imagine that someone qualifies for low in a loose eight handed game slightly more frequently than in a seven handed game - but probably still close to roughly three hands out of four.

Just my opinion. Never a guarantee I did the math correctly.

If anyone has a better value, simulated or calculated, I’d appreciate a correction.

Buzz

[Phat Mack]

I don't have access to my Dos box right now, so I couldn't run a Poker Probe sim, but here are the results from a Poker Calculator sim. Poker Calculator is a freeware program that executes with Java. The problem with it and some other simulators is that it doesn't tell you how many scoops took both high and low, and how many took high with no low qualifying....

<pre><font class="small">code:</font><hr> Monte carlo simulation results from Poker Calculator 1.1.2
7 Card Stud hi/lo 8/b, 100000 combinations tested.

Hand 1:
Random hand

Hand 2:
Random hand

Hand 3:
Random hand

Hand 4:
Random hand

Hand 5:
Random hand

Hand 6:
Random hand

Hand 7:
Random hand

Hand | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
------+--------+--------+--------+--------+--------+--------+--------+
High | 10189 | 10022 | 10119 | 10216 | 10053 | 10230 | 10173 |
Draw | 437 | 431 | 394 | 414 | 382 | 418 | 402 |
Lose | 75501 | 75486 | 75418 | 75380 | 75403 | 75236 | 75375 |
Scoop | 4047 | 4077 | 4109 | 4102 | 4250 | 4162 | 4114 |
Low | 9879 | 10040 | 10016 | 9938 | 9963 | 9998 | 9993 |
------+--------+--------+--------+--------+--------+--------+--------+
Win% | 14.19% | 14.22% | 14.28% | 14.28% | 14.35% | 14.38% | 14.3% |
------+--------+--------+--------+--------+--------+--------+--------+


Hand no.1:
High Card win: 0 draw: 0 lose: 17339
Pair win: 516 draw: 1 lose: 43147
Two Pair win: 4050 draw: 1 lose: 19606
Three of a Kind win: 2127 draw: 0 lose: 2832
Straight win: 2771 draw: 48 lose: 1805
Flush win: 2292 draw: 0 lose: 684
Full House win: 2289 draw: 0 lose: 301
Quads win: 159 draw: 0 lose: 0
Straight Flush win: 32 draw: 0 lose: 0


Hand no.2:
High Card win: 0 draw: 0 lose: 17565
Pair win: 494 draw: 0 lose: 43356
Two Pair win: 3971 draw: 1 lose: 19368
Three of a Kind win: 2014 draw: 0 lose: 2826
Straight win: 2741 draw: 39 lose: 1797
Flush win: 2280 draw: 1 lose: 696
Full House win: 2395 draw: 0 lose: 252
Quads win: 176 draw: 0 lose: 0
Straight Flush win: 28 draw: 0 lose: 0


Hand no.3:
High Card win: 0 draw: 0 lose: 17402
Pair win: 521 draw: 2 lose: 43130
Two Pair win: 4071 draw: 0 lose: 19648
Three of a Kind win: 2103 draw: 0 lose: 2733
Straight win: 2792 draw: 38 lose: 1821
Flush win: 2278 draw: 1 lose: 714
Full House win: 2266 draw: 0 lose: 282
Quads win: 166 draw: 0 lose: 1
Straight Flush win: 31 draw: 0 lose: 0


Hand no.4:
High Card win: 0 draw: 0 lose: 17432
Pair win: 547 draw: 1 lose: 43256
Two Pair win: 4065 draw: 2 lose: 19458
Three of a Kind win: 2081 draw: 0 lose: 2715
Straight win: 2802 draw: 41 lose: 1813
Flush win: 2255 draw: 0 lose: 674
Full House win: 2377 draw: 0 lose: 288
Quads win: 160 draw: 0 lose: 2
Straight Flush win: 31 draw: 0 lose: 0


Hand no.5:
High Card win: 0 draw: 0 lose: 17710
Pair win: 504 draw: 0 lose: 43125
Two Pair win: 4064 draw: 0 lose: 19370
Three of a Kind win: 2126 draw: 0 lose: 2715
Straight win: 2786 draw: 37 lose: 1762
Flush win: 2332 draw: 1 lose: 707
Full House win: 2288 draw: 0 lose: 269
Quads win: 179 draw: 0 lose: 1
Straight Flush win: 24 draw: 0 lose: 0


Hand no.6:
High Card win: 0 draw: 0 lose: 17344
Pair win: 541 draw: 0 lose: 43339
Two Pair win: 3988 draw: 0 lose: 19462
Three of a Kind win: 2061 draw: 0 lose: 2729
Straight win: 2820 draw: 34 lose: 1797
Flush win: 2315 draw: 1 lose: 672
Full House win: 2447 draw: 0 lose: 229
Quads win: 183 draw: 0 lose: 1
Straight Flush win: 37 draw: 0 lose: 0


Hand no.7:
High Card win: 0 draw: 0 lose: 17372
Pair win: 548 draw: 0 lose: 43293
Two Pair win: 3965 draw: 0 lose: 19582
Three of a Kind win: 2118 draw: 0 lose: 2706
Straight win: 2822 draw: 25 lose: 1759
Flush win: 2296 draw: 0 lose: 706
Full House win: 2289 draw: 0 lose: 268
Quads win: 211 draw: 0 lose: 2
Straight Flush win: 38 draw: 0 lose: 0
</pre><hr>

[Buzz]

Mack - Thanks. Very helpful!

We know from experience that sometimes we get quartered for low in stud-8, (although not nearly as much as in Omaha-8 where we are often sharing the same three low board cards with an opponent). I wonder if the low totals include pots where low was shared.

In any event, evidently my approximation of 75% for a seven handed game was off a bit. Using the numbers you supplied, 69827/100000 = about 70% is the maximum percentage of the time someone wins for low in a seven handed game.

Then, depending on how often quartering takes place for low in stud-8, and what the low data presented from the simulation represents, it seems we may need to revise that 70% downward.

"The problem with it and some other simulators is that it doesn't tell you how many scoops took both high and low, and how many took high with no low qualifying."

Don't all the scoops represent wins for both high and low? (Otherwise it would not make sense to record the highs and scoops separately).

My dilemma with the simulation data is different. I don’t know how many of the 69827 reported low wins represent a draw for low and how many don’t.

Thanks again for the data.

Buzz

[Buzz]

Mack - Wait! I see it!

The following differences must represent the low draws.
437_____431_____394_____414_____382_____418_____40 2
_50______41______41______44______38______35______2 5

Thus the low draws must be as follows:
387_____390_____353_____370_____344_____383_____37 7
Total low draws from simulation data must be 2604.

69827 - 2604 = 67223.

Thus someone has a qualifying low in a seven handed game of stud-8 about 67% of the time.

As an aside, when you make a winning low hand in a seven handed stud-8 game, according to Mack’s simulation data, evidently you are quartered about 3.8% of the time. (Interesting. Getting quartered for low in stud-8 occurs much less often in stud-8 than in Omaha-8. In a loose seven handed game of Omaha-8, an A-2 nut low figures to be quartered 31% of the time and sixthed another 3% of the time).

Darned if I didn’t turn this into an Omaha-8 post.

Buzz

[therazz]

Buzz &amp; Mack, thank you very much for efforts. It is greatly appreciated.

[Phat Mack]

Buzz, I don't get it. (My Chem Prof heard this a lot.) What are you subtracting from the draws?

[Buzz]

“Buzz, I don't get it. (My Chem Prof heard this a lot.) What are you subtracting from the draws?”

Mack - You provided data for seven “hands.” (Actually “hand,” as we’re using it here, might better be termed “position”). You listed wins, draws, and losses for High Card, Pair, Two Pair, Trips, Straight, Flush, Full House., Quads, and Straight Flush. These are obviously all listings for high (and not low). This data is very interesting data, and relates, I think, indirectly to lows.

I had my computer add together the total wins, draws, and losses for each hand. (I prefer using the word “ties” here instead of “draws,” because draw also means seeing another card, but I will continue to use the “draw” and “hand” terminology evidently provided by the simulator).

The totals are given below for each hand. This chart is one I developed from totaling data from charts you provided.

Chart I
Hand no....win.....draw.....lose.....
...1........14236.....50.....85714
...2........14099.....41.....85860
...3........14228.....41.....85731
...4........14318.....44.....85638
...5........14303.....38.....85659
...6........14392.....35.....85573
...7........14287.....25.....85688

You also provided another chart:

Chart II
Hand
|.....1....|.....2.....|....3.....|.....4....|.... .5.....|.....6....|.....7....|
|------+--------+--------+--------+--------+--------+--------+--------+
High
| 10189 | 10022 | 10119 | 10216 | 10053 | 10230 | 10173 |
Draw
|..437...|...431...|..394...|..414...|...382...|.. 418...|...402..|
Lose
| 75501 | 75486 | 75418 | 75380 | 75403 | 75236 | 75375 |
Scoop
| 4047...| 4077..| 4109...| 4102...| 4250...| 4162..| 4114.. |
Low
| 9879.. | 10040 | 10016 | 9938.. | 9963.. | 9998.. | 9993.. |

The total “draws” for each “hand” in one chart are listed vertically, and in the other horizontally, but you can easily see there is a huge discrepancy. The explanation must be that the draws in this last chart refer to both high and low.

Making that assumption, I subtracted the “draws” for high in Chart I
from the total “draws” for both high and low in Chart II to get the “draws” for low. (“Draws,” as I’m using the word here, means “ties”). Some of the ties for low could be due to sixthing, rather than quartering, but I suspect not many, since the frequency of ties for low is only 3.8%. That’s assuming I have correctly interpreted your data. Never a guarantee, but at least my interpretation makes sense to me, and I hope also makes sense now to you. If not, let me know and I’ll try to make my explanation clearer.

Buzz

[Phat Mack]

I see what you did. Ingenius! I never looked past the upermost chart. A minor quibble: I think you should have taken the number of ties for low, 2604, halved it to 1302 and then added it to the other lows for a total of 71129. JMO, let me think about it some more. I also note that the figures in what you call chart 2 above seem to have 50-something too many hands in them and this troubles me a little.