Estimating equity for five card PLO
Hi everyone,
In hold 'em the rule of 2 and 4 is used to roughly approximate equity. Do the smart people here have an idea of a similar "rule" for five card PLO in a full ring 8-handed game? Does this change when fewer people are dealt in?
We have five cards in our hand and let's say everyone is dealt in so 40 cards are dealt out of 52. Someone bets the pot. Assuming they have a strong made hand, what is an easy way to estimate our equity, either all in or to see one more card? As an example let's say we have a nut flush draw and nut open ender. We have seven non-board pairing flush outs and six straight outs, or 13 outs. But there are only 12 cards left in the deck.
Thanks,
DT
6 Replies
Cards in other players' hands are unseen to you, just as cards in the stub and muck are. All unseen cards are treated the same in the calculation so it doesn't matter how many players were dealt in.
Your chances of hitting a draw improve slightly the more cards you've seen, including your own hole cards, that would be bricks for that draw. For example: you're drawing to a flush on the flop. In holdem your 9 outs are among 47 unseen cards. But in 5 card omaha, if your other 3 hole cards don't block your draw, you have 9 outs among only 44 unseen cards, so your odds of hitting are a bit better.
IIRC the 2x and 4x rule of thumb for outs in holdem is a slight overestimate that gets worse as the number of outs increase. So it should be more accurate, maybe becoming a bit of an underestimate, for a game with more hole cards. I don't think it'd be worthwhile to try to come up with a different heuristic.
In holdem, the 2x 4x rule is simply an arithmetic approximation. After the flop there are 52-2-3 = 47 unknown cards left.
If you have N clean outs after the flop, then the probability you will hit by the river is
P =n/47 +(1-n/47)*n/46 = n/47 +n/46 - n^2/(46*47).
The approximation replaces the 46 and 47 with 50 and drops the last term assuming n is not too large, say 15 or less. Then
4x rule: P = n/50 + n/50 = 2n/50 = 4 * (n/100). So if n= 9, as you would have with a four flush, P = 36%.
2x rule: For the turn, P = n/ 46 ~ n/50 =2 *( n/100)
Note: There are several methods to make adjustments for a better approximation.
I’m not sure at what street you are posing in your question but the same estimating could be used, if the number of unseen cards is, say, greater than 46. But, since you hold 5 cards and the flop has 3, there are 52-8 = 44 unseen cards and therefore the 2x-4x approximation is not very good. Also, I would think that estimating the clean outs in PLO is generally a challenge.
In holdem, the 2x 4x rule is simply an arithmetic approximation. After the flop there are 52-2-3 = 47 unknown cards left.
If you have N clean outs after the flop, then the probability you will hit by the river is
P =n/47 +(1-n/47)*n/46 = n/47 +n/46 - n^2/(46*47).
The approximation replaces the 46 and
Thanks, so can we simply replace 47 and 46 with 44 and 43 in the above equations to come up with the solution?
For one card to come, the general equation for hitting an out on the river given the turn is
P (hit on river) = number of outs/number of remaining cards,
For hitting an out on the turn or river given the flop, the probability is
P(hit on turn) + P(miss turn)*P(hit on river given miss on the turn)
So, in PLO5 with 5 cards dealt to a player, the number of remaining cards in the deck is lower than in hold’em. As I previously mentioned, I think determining your outs is the challenge.
Thank you. Let’s just assume our outs are live.
Would you mind walking through an example? For instanceÂ…
LetÂ’s say we have an open ender to the nuts and the nut flush draw on the flop and we know our opponent has naked top set. We have seven clean flush outs and six additional straight outs.
So on the turn would it be 13/43 or around 30%? That seems about right. (I got 29% in an equity calculator.)
And on the flop it would be Â…?
On the flop, 5+3 = 8 cards have been dealt so there are 44 remaining in the deck. If you have 13 out,s the probability one falls on the turn is 13/44 for a hit. The probability one doesn’t fall is 31/44 and if that’s the case there are 43 remaining cards and a hit on the river has probability = 13/43.
On combining:
Pr(hit on turn or river (or both) = 13/44 + 31/44 * 13/43 = 50.8%
Another way: Pr(hit) = 1 - Pr(don’t hit on turn and river)
= 1 - 31/44 * 30/43 = 50.8% (31 “bad” cards on turn and 30 “bad” on the river)
On my Tumblr blog site, Hold 'em Mathology I have two blogs on equity estimation that provide other approaches that can possibly be modified for PLO5.
