Jackpot probabilities for 5-card Omaha Hi Lo

I am developing a jackpot scheme for a 5 card Omaha H/L game and would like to know the probability that the jackpot will be dealt on any given hand. Assuming that the game is full (8 players dealt in) I would like to know the probability of the jackpot being dealt if the qualification is set at four deuces beaten, four sixes beaten, four tens beaten, and straight flush beaten. Can someone walk me through the math on this?

Quote: TrishTheFish

I am developing a jackpot scheme for a 5 card Omaha H/L game and would like to know the probability that the jackpot will be dealt on any given hand. Assuming that the game is full (8 players dealt in) I would like to know the probability of the jackpot being dealt if the qualification is set at four deuces beaten, four sixes beaten, four tens beaten, and straight flush beaten. Can someone walk me through the math on this?

Does a pair have to be in the hole? for the jackpot? Suppose the board is AcAdAsJcJd, Player 1 has AhQh, and Player 2 has JhJc. Is this a bad beat?

No, the player is not required to hold a pair in hand, but the game is Omaha Hi/Lo, so the player must use two cards from their 5 card starting hand and three cards from the five on the board.

No, the player does not have to hold a pair in their hand, but they must comply with the rules of Omaha, by using two cards from their (5 card) hand and three cards from the (5 card) board.

Quote: Ayecarumba

Does a pair have to be in the hole? for the jackpot? Suppose the board is AcAdAsJcJd, Player 1 has AhQh, and Player 2 has JhJc. Is this a bad beat?

No, the player is not required to hold a pair in hand, but the game is Omaha Hi/Lo, so the player must use two cards from their 5 card starting hand and three cards from the five on the board.

Quote: TrishTheFish

No, the player is not required to hold a pair in hand, but the game is Omaha Hi/Lo, so the player must use two cards from their 5 card starting hand and three cards from the five on the board.

Need help from the Wizard (or other math inclined members) on this one. Here is the Wizard's formula for calculating the odds of a player getting a single four of a kind in traditional four card Omaha (https://wizardofodds.com/ask-the-wizard/156/):

(13*combin(4,2)*combin(48,2)*combin(2,2)*combin(46,3)-combin(13,2)*combin(4,2)*combin(4,2)*combin(2,2)*combin(2,2)*44)/(combin(52,4)*combin(48,5))

or 1 in 347.


Since the player's 2 cards are now chosen from 5 hole cards instead of 4, does that mean the correct formula is:

(13*COMBIN(5,2)*COMBIN(47,3)*COMBIN(2,2)*COMBIN(45,3)-COMBIN(13,2)*COMBIN(5,2)*COMBIN(5,2)*COMBIN(2,2)*COMBIN(2,2)*42)/(COMBIN(52,5)*COMBIN(47,5))

or 1 in 133?

Thanks for your help!

Quote: Ayecarumba

Need help from the Wizard (or other math inclined members) on this one. Here is the Wizard's formula for calculating the odds of a player getting a single four of a kind in traditional four card Omaha (https://wizardofodds.com/ask-the-wizard/156/):

(13*combin(4,2)*combin(48,2)*combin(2,2)*combin(46,3)-combin(13,2)*combin(4,2)*combin(4,2)*combin(2,2)*combin(2,2)*44)/(combin(52,4)*combin(48,5))

or 1 in 347.


Since the player's 2 cards are now chosen from 5 hole cards instead of 4, does that mean the correct formula is:

(13*COMBIN(5,2)*COMBIN(47,3)*COMBIN(2,2)*COMBIN(45,3)-COMBIN(13,2)*COMBIN(5,2)*COMBIN(5,2)*COMBIN(2,2)*COMBIN(2,2)*42)/(COMBIN(52,5)*COMBIN(47,5))

or 1 in 133?

Thanks for your help!

I'm refreshing this in hopes that someone can check out the calculation. 1 in 133 seems awfully low for making quads, even with five hole cards and a five card board.

Actually it's pretty close, but only because of compounding of errors. The Ask the Wizard thread only considers quads made from a pair in the hand. According to the original poster, making quads with trips on board will also count, so that should also be counted. You can't use Combin(5,2) in that formula for this version because Combin(4,2) was representing the 2 cards that you choose from a given rank to make the pair. The deck still has 4 cards per rank.

The wizard calculated the probabilities for Omaha here: https://wizardofodds.com/games/poker/

When trips on board are also included the correct probability for regular omaha is 1 in 208.32. The probability for 5cd omaha is about 1 in 131.6. I say about since I am having some minor double counting issues right now.


As for the original BBJ question, one can use PQL to estimate the relative chances that the BBJ will hit by ignoring folding:

http://www.propokertools.com/pql

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p8, river) = FALSE) ) as quad_2s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_2S_BEAT
355326 262 (0.07%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p8, river) = FALSE) ) as quad_6s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_6S_BEAT
274942 140 (0.05%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p8, river) = FALSE) ) as quad_10s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_10S_BEAT
379902 83 (0.02%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p8, river) = FALSE) ) as str8_flush_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials STR8_FLUSH_BEAT
381438 39 (0.01%)


Again, remember, this assumes no folding at any point. Folding makes these results even less likely.

deleted

OP refers to "5-card omaha". I am not familiar with the game, but I assume that it is like Omaha, only, you have a 5 card hand instead of 4.

Quote: AxiomOfChoice

OP refers to "5-card omaha". I am not familiar with the game, but I assume that it is like Omaha, only, you have a 5 card hand instead of 4.

This is correct. And is often played as hi/lo with an 8 or lower qualifier. The 5-card hi/lo8 game is often referred to as "Big O".

Quote: tringlomane

Actually it's pretty close, but only because of compounding of errors. The Ask the Wizard thread only considers quads made from a pair in the hand. According to the original poster, making quads with trips on board will also count, so that should also be counted. You can't use Combin(5,2) in that formula for this version because Combin(4,2) was representing the 2 cards that you choose from a given rank to make the pair. The deck still has 4 cards per rank.

The wizard calculated the probabilities for Omaha here: https://wizardofodds.com/games/poker/

When trips on board are also included the correct probability for regular omaha is 1 in 208.32. The probability for 5cd omaha is about 1 in 131.6. I say about since I am having some minor double counting issues right now.


As for the original BBJ question, one can use PQL to estimate the relative chances that the BBJ will hit by ignoring folding:

http://www.propokertools.com/pql

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("2c2d2h2s3s") AND bestHiRating(p8, river) = FALSE) ) as quad_2s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_2S_BEAT
355326 262 (0.07%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("6c6d6h6s2s") AND bestHiRating(p8, river) = FALSE) ) as quad_6s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_6S_BEAT
274942 140 (0.05%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("TcTdThTs2s") AND bestHiRating(p8, river) = FALSE) ) as quad_10s_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUAD_10S_BEAT
379902 83 (0.02%)

PQL Query:
select count((hiRating(p1, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p1, river) = FALSE) OR (hiRating(p2, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p2, river) = FALSE) OR (hiRating(p3, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p3, river) = FALSE) OR (hiRating(p4, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p4, river) = FALSE) OR (hiRating(p5, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p5, river) = FALSE) OR (hiRating(p6, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p6, river) = FALSE) OR (hiRating(p7, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p7, river) = FALSE) OR (hiRating(p8, river) >= rateHiHand("As2s3s4s5s") AND bestHiRating(p8, river) = FALSE) ) as str8_flush_beat
from game="omaha85", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials STR8_FLUSH_BEAT
381438 39 (0.01%)


Again, remember, this assumes no folding at any point. Folding makes these results even less likely.

Thanks tringlomane! Is there any way to estimate the effect on these totals of players folding? I assume that it only makes the odds of a bad beat even rarer.

Quote: Ayecarumba

Thanks tringlomane! Is there any way to estimate the effect on these totals of players folding? I assume that it only makes the odds of a bad beat even rarer.

There is, but unfortunately the free code doesn't allow you to run the simulator long enough to use precentile preflop ranges. :(