Hold'Em: Nine Handed Odds vs Ten Handed Odds

Hello everyone!

Just wanted to know the general probability differential between nine handed and ten handed poker.

Specifically for Bad Beats, but was curious as to if the difference for any hand calculation is basically more or less the "triangular" ratio of the possibility?

Meaning if the chances of 2 people having a hand is 1 in X, then the chances of 2 people having a hand is 3 in X when playing 3 handed?

In turn meaning:
2 players = 1 in X
3 players = 3 in X
4 players = 6 in X
5 players = 10 in X
6 players = 15 in X
7 players = 21 in X
8 players = 28 in X
9 players = 36 in X
T players = 45 in X

Thus any event on a nine handed table would happen approximately 4/5th as often as on a ten handed (36/45)?

I'm guessing I've left out like the possibility of a 3rd player getting a bad beat hand as well. But I'm thinking I've really over simplified it too much.

So in a 9 handed poker room would the type 3 bad beat jackpot with
minimum Quad 8s (all pockets must play, and MUST be pocket pairs for quads) be:

0.00000519 * 36 / 45 = 0.00000415

Essentially 1 in 240,963 instead of 1 in 192,678?
Or is my approach completely wrong? lol


(Note .. I got the 0.00000519 from Wizard of Odds)


PS .. Thanks for having great sites and all the work behind all the information you share! :)

As long as the other players do not reveal the contents of their hands, the odds don't change regardless of the number of players.

https://wizardofodds.com/games/texas-hold-em/bad-beat-jackpots/

Quote: DJTeddyBear

As long as the other players do not reveal the contents of their hands, the odds don't change regardless of the number of players.

https://wizardofodds.com/games/texas-hold-em/bad-beat-jackpots/

No he's right. Playing at a table with only 2 players dealt cards it's roughly 45 times less likely to occur.

And to make things even worse all these numbers the Wizard computed assume no one ever folds. So even at a 10 handed table, the actual probability of triggering the bad beat jackpot is significantly lower.

Are you saying the odds of a bad beat coming out are the same regardless as to if there are 9 players or 10 players who get cards at the start of a hand?

I'm pretty sure that can't be true. Think about it .. sometimes one of the two hands involved will come out on for the 10th player .. which would not have happened if it was nine-handed .. thus taking away some of the odds.

What you're talking about is single hand combinations .. like the fact you as a player will get a royal flush just as often as you play head to head as you will if there were 10 players. That's true .. but I'm pretty sure it isn't when the "event" in question relies on multiple hands as is te case with a bad beat jackpot (which involves at least 2 hands).

Think about this deliberately obvious example .. what are the chances of having two Tails on coin flips .. it's going to be different if you have 2 coins vs 10 coins .. although the argument still holds true that your odds will always be 50% for each specific coin flip .. but what I'm referring to is different.

Yeah .. thanks Tringlomane .. do you know if that's essentially the exact ratio for all similar comparisons?

" I'm pretty sure that can't be true. Think about it .. sometimes one of the two hands involved will come out on for the 10th player .. which would not have happened if it was nine-handed .. thus taking away some of the odds."

You are right ! Might happen more in 9 handed games. One less maniac to drive out the hand that might be involved in a Bad Beat .

Quote: MrLeft

Yeah .. thanks Tringlomane .. do you know if that's essentially the exact ratio for all similar comparisons?

Roughly speaking, I think it is. It wouldn't be exact though because as you initially pointed out, with more players you have the small chance of either 3 or even 4 players qualifying for a "bad beat", but that occurrence would only be counted one time. To be totally precise, you can't even use the method the wizard used. But Brian Alspach, who I consider to be a "wizard" at poker math, did it for the Party Poker Online Bad Beat back in late 2004. Took him 20 pages, and is very complex to understand. I've never tried to follow every little step.

http://people.math.sfu.ca/~alspach/comp46.pdf

Hopefully Brian and his wife are enjoying his retirement in Australia, where he moved a few years ago.

Nice guys, even answers email from fools, like me.

lol .. I found that in my google search .. admittedly gave up somewhere in the middle of the second paragraph because I had no clue even what the terms were that he was using .. nevermind the math! lol

I'm curious if a 3+ player Quad+ bad beat has ever been dealt in real world hold'em. I came about as close as possible a few years ago .. but it only turned out to be 2 players .. lol.

Is there any established guestimate of the average percentage of bad beat hands folded in real-world casino live games? Or even in online cash games where I assume that stat could really be collected?

Quote: MrLeft

lol .. I found that in my google search .. admittedly gave up somewhere in the middle of the second paragraph because I had no clue even what the terms were that he was using .. nevermind the math! lol

I'm curious if a 3+ player Quad+ bad beat has ever been dealt in real world hold'em. I came about as close as possible a few years ago .. but it only turned out to be 2 players .. lol.

Is there any established guestimate of the average percentage of bad beat hands folded in real-world casino live games? Or even in online cash games where I assume that stat could really be collected?

It makes a significant difference. One of the guys that worked on pro poker tools estimated that in a reasonably loose PLO game, folding can lower the probability of it being hit by almost a factor of 10.

http://forumserver.twoplustwo.com/25/probability/club-running-bad-beat-jackpot-set-correctly-plo-players-1071511/

He used this program, which he helped create:

http://www.propokertools.com/pql

Cool .. thanks for the links .. the first one you pointed to was more about actual odds .. The factor of 10 comment was more changing the BB requirement from Quad A down to Quad 2.

but on it I found a link to this which is more what we're talking about ...
http://forumserver.twoplustwo.com/25/probability/what-point-does-bad-beat-jackpot-become-ev-961733/

There it someone did crude math to point the number more around 50% .. That number obviously shifts the higher the jackpot goes and more people play "loto-poker" style .. lol

In Montreal our bad beat jackpot is at a record $660,000 .. had to add more tables!

Quote: MrLeft

Cool .. thanks for the links .. the first one you pointed to was more about actual odds .. The factor of 10 comment was more changing the BB requirement from Quad A down to Quad 2.

but on it I found a link to this which is more what we're talking about ...
http://forumserver.twoplustwo.com/25/probability/what-point-does-bad-beat-jackpot-become-ev-961733/

There it someone did crude math to point the number more around 50% .. That number obviously shifts the higher the jackpot goes and more people play "loto-poker" style .. lol

In Montreal our bad beat jackpot is at a record $660,000 .. had to add more tables!

Thanks for finding a thread I did the math in...haha. But as for the omaha thing, "bachfan" discussed preflop folding and lowering it to quad deuces in two different posts, and they both dropped things more significantly than 50%. But maybe that's because omaha has four cards vs two.

http://forumserver.twoplustwo.com/showpost.php?p=28317713&postcount=20