omaha8 calculations

[nanoking]

I was messing around with Excel. When I play online I have a spreadsheet open with a table of outs and odds required to call flop/turn/both (from Sklansky's tables). Normally I play holdem, but I found myself playing an omaha8 single table tourney since the holdem ones were filling up too fast for me to get in (customer support says the lobby refreshes every 20 seconds...grrr).

Anyway, I redid the tables for Omaha since you see more cards. Doing that got me reviewing omaha differences: winning hand strength goes up vs. holdem, can get counterfeited, need to accurately determine if hand drawing for will actually win, need to draw to or have the nuts, etc.

I needed to consider split pots. I added 3/4, 1/2, 1/4 columns in my table so that the odds values would be for the portion of pot I'd actually win.

I also needed to consider scooped pots. I added a section that gave the overall pot odds you needed to bet given X high outs and Y low outs each to half the pot. I did this using an expected value formula for the 4 cases: win high/win low, win high/lose low, lose high/win low, lose high/lose low. The breakeven pot odds were different from the average of the high and low odds.

Sorry if these musings have been posted before. First time I have considered them. It's nice while playing online to be able to know exactly the breakeven pot odds for any given high and low odds.

Example so people can check my math:

on turn
high outs 7 high odds 10.86 ph 0.084
low outs 16 low odds 3.63 pl 0.22
average 7.24
total odds 6.53

on river
high outs 7 high odds 10.57 ph 0.086
low outs 16 low odds 3.5 pl 0.22
average 7.04
total odds 6.36

turn/river
high outs 7 high odds 4.90 ph 0.17
low outs 16 low odds 1.39 pl 0.42
average 3.14
total odds 3.16

Accurate? Interesting? Useless?

-Nanoking

[Buzz]

"on turn
high outs 7 high odds 10.86 ph 0.084
low outs 16 low odds 3.63 pl 0.22
average 7.24
total odds 6.53"

nanoking - Let's say you hold AA23 non-suited and the flop is AhKh8h. In that case you'd have 7 outs for high and 16 outs for low. The odds against your making a full house/quads on the turn would be 38 to 7 against, or about 5.4 to 1 against. The probability of your making a full house/quads on the turn would be 7/45 = 0.156. I don't see either of these reflected anywhere in your numbers.

Your "average," (10.86+3.63)/2 = 7.245, doesn't make any sense at all.

Just my opinion.

Buzz

[Phat Mack]

I think it's an interesting project, but I don't follow the math. I also don't understand how you came up with the averaging.

I like Buzz's method of using specific hands to illustrate your example. Some additional food for thought: if you hit your high hand on the turn, what is the probability a low hand won't make on the river? If you hit your low hand on the turn, what is the probability it will still be nut low on the river?

[nanoking]

Corrections and followup:

If you are going for high or low and expect to only win half the pot you have to use half the pot for odds calculations. So if you have 7 outs that would give you odds of 5.43 to win a half pot. So the actual pot would have to be 10.86 times larger than your bet.

BUT I did make a couple errors in my overall odds formula. I was using probability = 1/(odds+1) with the doubled odds (10.86 instead of 5.43). I also had a small typo in my EV formual in my spreadsheet

On the turn, With 7 outs towards half the pot your probability of winning half the pot is 7/45 or 0.156.

Overall the equations are:
Odds with X outs to see turn: 45/X - 1
Odds with X outs to see river: 44/X - 1
Odds with X outs for both turn/river: 1980/(89*X-X^2)-1
probabilities = X/45, X/44, (89*X-X^2)/1980

EV = sum(probabilities*payouts)
set EV = 0 and solve for breakeven overall odds

after simplifying, overall odds = (2-ph-pl)/(ph+pl)
(somebody doublecheck THIS one, it's getting late /images/graemlins/smile.gif)

Example for high outs = 7, low outs = 16:

ph 0.156
pl 0.356
total odds 2.913

ph 0.159
pl 0.364
total odds 2.826

ph 0.290
pl 0.590
total odds 1.273

Much better! (and now it fits Buzz's example hand better)

[Buzz]

"So if you have 7 outs that would give you odds of 5.43 to win a half pot."

Nanoking - Usually one sees odds written as "5.43 to 1" rather than "5.43." Not a big deal.

"So the actual pot would have to be 10.86 times larger than your bet."

O.K. I follow.

"Odds with X outs to see turn: 45/X - 1"

O.K. I follow. Those are the odds against making your hand on the turn

"Odds with X outs to see river: 44/X - 1"

I follow.

"Odds with X outs for both turn/river: 1980/(89*X-X^2)-1"

Whoa. Let's check it.
3.45-1 = 2.45 for 7 outs.
Interesting. Your method seems to work.

Usual method with 7 outs: 1-(38/45)*(37/44) = 0.29 = probability.
Odds are therefore 2.45 to 1.

Another way:
(7*6/2+7*38)/990 = 287/990 = 0.29 = probability.

"after simplifying, overall odds = (2-ph-pl)/(ph+pl)"

What do ph and pl represent?

Buzz

[nanoking]

ph and pl are the probabilities of making your high and low hands respectively.

I want to know the overall pots odds taking into consideration my high and low outs together.

To get that overall odds formula I used the following outcomes:

win high, lose low leads to winning: pot/2 - bet/2
lose high, win low " ": pot/2 - bet/2
win high, win low: pot
lose high, lose low: -bet

[nanoking]

Or should I be using EV = sum(probabilities*PROFITS)? I think I should be...

profits for outcomes:

win/win: total pot - bet
win/lose: half pot - bet
lose/win: half pot - bet
lose/lose: -bet

In which case I get overall odds = 2/(ph+pl) which is just 1/average

Then the "scoop-equivalent" outs (for turn and river at least) = cards remaining/(overall odds+1)

Trying again (for 7 high outs and 16 low outs):

Turn:
ph 0.156
pl 0.356
total odds 3.913
scoop equiv outs 9.159
scoop fraction 0.398

River
ph 0.159
pl 0.364
total odds 3.826
scoop equiv outs 9.117
scoop fraction 0.396

Both
ph 0.290
pl 0.590
total odds 1.695
scoop equiv outs 9.207
scoop fraction 0.400

SO! Just adding up your outs for the high half and the low half of the pot, multiplying by 0.4, and calculating pot odds using the whole pot would just as good as this rigamarole.

-NK

p.s. thanks to Buzz for the term "scoop-equivalent"