Alright I finally got a hold of Pokerstove and have been playing with some numbers. Here's what I found:
I had to take some liberty in making assumptions for our opponents range of hands...sorry if you disagree.
PREFLOP--Opponents range of hand:
Any pocket pair
Any suited connectors (45 and above--throws away 23s etc)
Any one-gap suited connectors (46 and above),
AQ+ (AK, AQ)
FLOP: 7[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/heart.gif[/img] 9[img]/images/graemlins/heart.gif[/img]
We could discuss the implications of this, but it gets much more interesting if we limit the range of our opponents hand after the flop betting.
Action: BB checks. Hero bets t140. BB raises to t300. The pot has $610.
TURN--Opponents range of hand
Any pocket pair 55+ (55,66,77, etc),
Any flush draw
Any straight draw
Any made straight (68)
Any pair with a gut draw (56s,67s,78s,89s,9Ts...not hearts)
Any 9Js (not hearts).
Any AK
Any AQs
I left in AK & AQs (only if suited) because that would mean that the opponent bluffed with 10% of his hands, which is what Harrington says is typical.
TURN POT EQUITY--Opponents Limited Range:
Hero's Equity= 75% of the pot
Villian's Equity= 25% of the pot
Then I figured the pot equity for the Hero and Villian based upon different River cards:
POT EQUITY--Based on various River cards and Villian's limited range of hands as stated above:
River: 2[img]/images/graemlins/spade.gif[/img]
Hero's Equity= 95%
Villian's Equity= 5%
River: T[img]/images/graemlins/spade.gif[/img]
Hero's Equity= 83%
Villian's Equity= 17%
River: J[img]/images/graemlins/spade.gif[/img]
Hero's Equity= 73%
Villian's Equity= 27%
River: 2[img]/images/graemlins/heart.gif[/img]
Hero's Equity= 86%
Villian's Equity= 14%
River: J[img]/images/graemlins/heart.gif[/img]
Hero's Equity= 68%
Villian's Equity= 32%
River: 6[img]/images/graemlins/spade.gif[/img]
Hero's Equity= 57%
Villian's Equity= 43%
River: 8[img]/images/graemlins/club.gif[/img]
Hero's Equity= 52%
Villian's Equity= 48%
River: 6[img]/images/graemlins/heart.gif[/img]
Hero's Equity= 50%
Villian's Equity= 50%
Below Hero has less equity:
River: 9[img]/images/graemlins/club.gif[/img]
Hero's Equity= 46%
Villian's Equity= 54%
River: 8[img]/images/graemlins/heart.gif[/img]
Hero's Equity= 46%
Villian's Equity= 54%
River: Q[img]/images/graemlins/club.gif[/img]
Hero's Equity= 42%
Villian's Equity= 58%
River: Q[img]/images/graemlins/heart.gif[/img]
Hero's Equity= 37%
Villian's Equity= 63%
Thoughts: It was more or less what I expected, but I thought that it was interesting that the 9[img]/images/graemlins/club.gif[/img] reduced the Hero's equity almost as much as an 8[img]/images/graemlins/heart.gif[/img]. I guess being counterfeited has a large influence.
Implications: So I grouped every possible card that could come on the river into a set based upon how significantly it effected Hero's pot equity.
GROUP A: River 2[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] 4[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] 7[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/heart.gif[/img] 5[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/heart.gif[/img] for a total of 10 cards.
Hero's Equity= 95%
GROUP B: River: A[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img]
(in other words, without a heart) for a total of 6 cards.
Hero's Equity= 89% of the pot.
GROUP C: River: T[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] 3[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] (in other words, w/out a heart) A[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/heart.gif[/img] 4[img]/images/graemlins/heart.gif[/img] 2[img]/images/graemlins/heart.gif[/img] for a total of 10 cards.
Hero's Equity= 83-86% of the pot.
GROUP D: River: J[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] [img]/images/graemlins/heart.gif[/img] T[img]/images/graemlins/heart.gif[/img] 3[img]/images/graemlins/heart.gif[/img] for a total of 6 cards.
Hero's Equity= 68-73% of the pot.
GROUP E: River: 6[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] 8[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] 6[img]/images/graemlins/heart.gif[/img] for a total of 7 cards.
Hero's Equity= 50-57% of the pot.
GROUP F: River: 9[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/spade.gif[/img] Q[img]/images/graemlins/club.gif[/img] [img]/images/graemlins/diamond.gif[/img] 8[img]/images/graemlins/heart.gif[/img] for a total of 6 cards.
Hero's Equity= 42-46% of the pot.
GROUP G: River: Q[img]/images/graemlins/heart.gif[/img] for a total of 1 card.
Hero's Equity= 37%
SO WHAT DOES ALL THIS MEAN?:
Note that the above groupings account for all 47 unseen river cards.
The pot has $770
On the River (assuming no betting):
Group A: Hero's Pot Equity 22% of the time will be $732 (95% of the pot)=$161
Group B: Hero's Pot Equity 13% of the time will be $685 (89% of the pot)=89
Group C: Hero's Pot Equity 22% of the time will be $651 (84.5% of the pot)=$143
Group D: Hero's Pot Equity 13% of the time will be $562 (73% of the pot)=$73
Group E: Hero's Pot Equity 15% of the time will be $416 (54% of the pot)=$62
Group F: Hero's Pot Equity 13% of the time will be $346 (45% of the pot)=$45
Group G: Hero's Pot Equity 2% of the time will be $285 (37% of the pot)=$6
RIVER POT EQUITY:
The weighted sum of the Hero's Pot Equity on the River based upon the opponents range of hands w/out future betting:
Pot Equity = (.22x732)+(.13x685)+(.22x651)+(.13x562)+(.15x416)+ (.13x346)+(.02x285)
= $579
RIVER POT EQUITY= $579 or 75% of the pot
Note that this is the exact same number that Poker Stove gave for our pot equity at the top of this post.
So how can you and I use this?
Dicuss what you'd do differently if a card from Group A vs Group B vs Group C vs Group D vs Group E came on the River. And you can discuss it in terms of how it effects the Hero's pot equity!!
What amount to bet on the Turn?
If we check the turn, what amount must we induce from the villian which will not risk our equity?
Or discuss why my range of hands for the Villian sucked and why I suck and why this post was a big waste of time. Or if someone wants to start a post about what the Villian's range is, and there is some consensus then I'd be happy to use it.
I made a similar analysis for the turn card about an hour ago. Here is the link:
Turn Card Pot Equity
I hope this is helpful.