Tuesday, February 21, 2006

Know thy Equity

(Most of this was originally posted over at Kickered, but I thought I'd compile it into a larger post here.)

I've been spending more time of late with PokerStove, which is a great, free application that calculates equity for assorted hands. It's been pretty useful, especially for certain situations where my gut read is actually pretty far off of the reality of the cold, hard math. I'd definitely recommend downloading and playing around with it, as even just plugging in random hands can be helpful.

Below you'll find some reasonably common situations, looking at different equity situations.

Know thy Equity #1: AA versus 3 random hands

Not really much comfort, when a pack of wild lemurs drags down your pretty AA, but something to keep in mind:

equity (%) win (%) tie (%)
Hand 1: 61.3701 % 61.22% 00.16% { AcAd }
Hand 2: 08.9116 % 08.70% 00.22% { random }
Hand 3: 16.2508 % 15.89% 00.37% { random }
Hand 4: 13.4674 % 12.93% 00.54% { random }

Even against 3 other truly random hands, your aces are still just 61% preflop.

And while lemurs may play lots of cards, they're usually at least a tiny bit selective, so it's even worse against a normal distribution of lemur hands:

equity (%) win (%) tie (%)
Hand 1: 57.9679 % 57.93% 00.03% { AcAd }
Hand 2: 15.6935 % 15.66% 00.03% { Jh5h }
Hand 3: 12.5280 % 12.49% 00.03% { QsTc }
Hand 4: 13.8105 % 13.78% 00.03% { 9d8h}

Moral of the story: Pocket aces are obviously the shizzle, but temper your expectations, especially in a crowded field, and don't fall in love with what's in reality just a 61% favorite versus 3 completely random hands.

Know Thy Equity #2: Hero has Top 2 Pair vs. OESD/Flush Draw vs. Bottom Set

This basic situation is where the hero flops top two pair, while one villain has an OESD+flush draw, while a second villain flops bottom set.

Board: Kc Qh 5c

equity (%) win (%) tie (%)
Hand 1: 17.2757 % 17.28% 00.00% { KdQd }
Hand 2: 44.1860 % 44.19% 00.00% { JcTc }
Hand 3: 38.5382 % 38.54% 00.00% { 5h5s }

This one always surprises me, as far as the fact that the OESD/flush draw is the favorite on the flop.

If you remove the second villain with the set, the hero with top 2 pair is still just roughly 51% to win on the flop, versus 49% for the villain with the big draws.

Moral of the Story: Don't underestimate the value of big drawing hands, even if you're fairly certain someone already has a big made hand.

Know Thy Equity #3: Gutshots Are Substantially Accretive to Equity

This came up in a MTT I played a few weekends back, and I thought it a bit illuminating.

Basic situation is that my aces got cracked by Q6s, who pushed all-in on a flop of Ks Td 9s. I had a sizable stack at the time, so it didn't really hurt me, but I thought it was interesting to run the numbers, as far as the added equity the gutshot adds to his hand.

Here are the numbers for the actual hand itself:

Board: Ks Td 9s

equity (%) win (%) tie (%)
Hand 1: 53.1313 % 53.13% 00.00% { AcAd }
Hand 2: 46.8687 % 46.87% 00.00% { Qs6s }

and then here are the numbers removing the gutshot draw, but maintaining the flush draw:

Board: Ks Td 2s
equity (%) win (%) tie (%)
Hand 1: 62.2222 % 62.22% 00.00% { AcAd }
Hand 2: 37.7778 % 37.78% 00.00% { Qs6s }

I'm sure this is old hat and very apparent to more knowledgable folks, but I wouldn't have guessed that the gutshot added quite that much equity, ~9%, to his hand.

Moral of the Story: Don't be too quick to label someone a lemur if they shove into you with a flush draw + gutshot, as they're often not as far behind as you might think.


Alan said...

Interesting. Great post.

Your odds may go down with each player added against AA, but your preflop EV goes up. At least until you add the 10th player.

A gutshot gives you 4 outs. I this is basically 1 to 10 odds. I need 10-to-1 pot odds before I'll stay in a hand with a gutshot. Very rarely will I be offered that.

But 1-to-10 is still 9%. Add a second card and you are looking at around 18%. That is nearly 1-to-4 odds to chase a gutshot from flop to river.

The flush draw changes the straight outs 3. That comes out to a 13%. Subtract your re-outs and I'm sure you get the 9%.

A flush draw with an open-straight draw with two cards gives you 15. That is around 64-54% depending on how double draws affect you (like a baby flush draw wouldn't want two suited cards to come out). That is a good thing to know when you are considering a jam on the flop.

jjok said...

VERY insightful....

Donkeypuncher said...

Very well done, sir.

Garthmeister J. said...

Yay Math!

Sparky said...

Ouch, my head...

The Litvak said...

Excellent post!
Now, if I can learn to stop pushing with Q6s the rest of the time...

StudioGlyphic said...

What Alan said regarding the gutshot...