Poker Odds - Texas Hold'em vs. 5-Card Omaha

Got into a heated discussion with a fellow poker player regarding poker odds; specifically, is the probability of making jacks full of anything or better in a 9-handed Texas Hold'em game higher or lower than the probability of making Aces full of anything or better in an 8-handed 5-card Omaha game? The topic came up due to the other player questioning the validity of a promotion at a poker room in which players in Hold'em games receive a drawing ticket for making hands of jacks full of anything or better while 5-card Omaha players have to make Aces full of anything or better to be eligible to receive a drawing ticket. Any help would be appreciated and if someone could figure out the specific odds for each game that would be really awesome.

Somebody with better math skills may be able to answer the question more adequately, but I believe that the probability of making a hand is irrelevant to the number of players or whether it's Texas Hold-Em or Omaha.

Usually, the reason for requiring a different hand has to do with how often hands are played. Also it will depend on whether or not, if Texas Hold-Em, both of the hole cards have to play in order to qualify for the promotion, as is usually the case in Bad Beat Jackpots. In Omaha hi-low, people are very likely to play any ace with any one other low card. As such, there's more likelihood that the player will luck into an Ace full house. On the flip side, a pair of Jacks in Omaha hi-low, without any supporting cards, is a poor hand, while in Texas Hold-Em, it's pretty good. (I don't know if the Omaha game you are speaking of is high-low or high only.)

My guess is that whoever is in charge of the poker room promotions perceives, perhaps correctly, that more people play more hands in Omaha than in Hold-Em. As such, upping the requirement to enter the drawing helps to "even out" the number of people who will be put into the drawing from the various tables.

What is 5 card omaha? If it really is omaha played with 5 hole cards, I'm pretty sure you are going to see bigger hands in that game than in THE. Probably easier to get the Aces full in 4 card Omaha

For Texas Hold'em do both hole cards need to be used to qualify? If so, it's definitely 5 card Omaha being easier. Even if both cards aren't needed for Hold'em, 5 card Omaha is still easier to hit at least in a no-folding scenario.

Texas Holdem

PQL Query:
select count(hiRating(p1, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p2, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p3, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p4, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p5, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p6, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p7, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p8, river) >= rateHiHand("JsJhJc2h2c") OR hiRating(p9, river) >= rateHiHand("JsJhJc2h2c") ) as jacks_full_or_better
from game="holdem", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*", p9="*"
Results:
Trials JACKS_FULL_OR_BETTER
600000 39169 (6.53%)

5-card Omaha

PQL Query:
select count(hiRating(p1, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p2, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p3, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p4, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p5, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p6, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p7, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p8, river) >= rateHiHand("AsAhAc2h2c") ) as aces_full_or_better
from game="omahahi5", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials ACES_FULL_OR_BETTER
254718 29389 (11.54%)

It's still slightly more likely in 4 card Omaha as well:

PQL Query:
select count(hiRating(p1, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p2, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p3, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p4, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p5, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p6, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p7, river) >= rateHiHand("AsAhAc2h2c") OR hiRating(p8, river) >= rateHiHand("AsAhAc2h2c") ) as aces_full_or_better
from game="omahahi", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials ACES_FULL_OR_BETTER
600000 46761 (7.79%)

For 5 card Omaha at least, quads may be a more equal qualifier to Jacks full in Holdem with using any number of hole cards (but this again assumes no folding).

PQL Query:
select count(hiRating(p1, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p2, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p3, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p4, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p5, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p6, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p7, river) >= rateHiHand("2c2d2h2s3s") OR hiRating(p8, river) >= rateHiHand("2c2d2h2s3s") ) as quads_or_better
from game="omahahi5", p1="*", p2="*", p3="*", p4="*", p5="*", p6="*", p7="*", p8="*"
Results:
Trials QUADS_OR_BETTER
397822 27075 (6.81%)

WOW!!! Awesome analysis. In hold'em both cards do not have to play. In fact, players can play the board to qualify to receive a promotional ticket. Does that change the probbability significantly?

5-card is played the same as 4-card with an extra hole card. Some houses run it 9-handed at the button is dealt out unless a player is away from the table. Other houses run it as an 8-handed game with no one being dealt out.

Good points. The player I was debating with stated he believed it was unfair to the Hold'em players to allow the Omaha hi/low players to be eligible for the promotion because it was easier to make qualifying hands (Aces full or better) in Omaha hi/low than in Hold'em (Jacks full or better).

Quote: CaliPokerBoss

WOW!!! Awesome analysis. In hold'em both cards do not have to play. In fact, players can play the board to qualify to receive a promotional ticket. Does that change the probbability significantly?

Sorry, I missed this earlier; Thanksgiving weekend and such. It definitely would change the probability, but the results I computed allow the hold'em players use 0, 1, or 2 cards to make their hand while for Omaha/5-cd Omaha both have to use exactly 2 cards from their hand. So the probabilities listed fit your promo's rules, but doesn't include the effect of folding before showdown. The more people fold, the less likely it will be hit. If you are looking to alter things to make it more equal, move the 5-cd Omaha requirement to quads. But in my opinion, it is also fine as is since it may entice people to play different games. I personally hate NLHE.