Estimating Chance You Are Beat On The Flop

[gaming_mouse]

I'm playing around with an idea for estimating the chance that you are behind on the flop. This idea may not be new, but I just thought of it and would like some feedback.

To start with a simplified version of the idea, let's assume you are at a uniformly tight table. Ignoring position for now, let's assume we can approximate our opps playable preflop hands using those suggested in HEFAP (meaning our opps play by that book, so to speak).

Here is a breakdown of those hands by card:

Total Playable Hands: 642 (out of all 1326 possible)

Percent Playable Hands Containing a 2: 0.03426791277258567
Percent Playable Hands Containing a 3: 0.040498442367601244
Percent Playable Hands Containing a 4: 0.0778816199376947
Percent Playable Hands Containing a 5: 0.102803738317757
Percent Playable Hands Containing a 6: 0.102803738317757
Percent Playable Hands Containing a 7: 0.11526479750778816
Percent Playable Hands Containing a 8: 0.1588785046728972
Percent Playable Hands Containing a 9: 0.21495327102803738
Percent Playable Hands Containing a T: 0.20249221183800623
Percent Playable Hands Containing a J: 0.22118380062305296
Percent Playable Hands Containing a Q: 0.17133956386292834
Percent Playable Hands Containing a K: 0.2087227414330218
Percent Playable Hands Containing a A: 0.22741433021806853

Let's assume also that the above counting and arithmetic is correct for now even though I might have slipped up. Now, to illustrate my idea, let's say you hold:

9 [img]/images/graemlins/heart.gif[/img] 9 [img]/images/graemlins/diamond.gif[/img]

Three opps see the following flop:

2 [img]/images/graemlins/spade.gif[/img] 5 [img]/images/graemlins/club.gif[/img] Q [img]/images/graemlins/heart.gif[/img]

Ignoring the possibility of sets and two-pairs for now, we want to estimate the chance that someone holds the queen. Each of our opps has a 17% chance of holding a Q. So the chance of at least one of them holding a Q is 1 - (.83)^3 = .42

This estimate is rough since opp position or preflop action may have made Q hands either more or less likely than many other playable hands. I don't know how much these considerations would alter the estimate, but I think the above should serve as a decent baseline.

How useful is such an estimate?

If it were useful, it would be easy to construct tables like the above for extremley loose players (who play any A for example) as well as extremely tight players.

I look forward to any comments/critiques.

gm

[kuro]

The problem with something like this is that it makes an assumption that everyone is playing the same starting hands. This just isn't the case.

Starting hand selection varies by player, position, stack size, number of players in the pot, size of the blinds, table atmosphere, etc.

This complexity makes your model pretty unusable. Then again it also is what makes poker so beautiful.

[Piers]

A goodish sort of thing to think about.But try playing with Poker calculator or pokerStove.

[jtr]

Hey, mouse.

I sympathize with your efforts here, but I suspect that if you build a simulation or work with an existing tool like PokerStove, you'll end up concluding that the collective wisdom that culminates in a book like SSHE is pretty much on the money.

Personally I worked for a while with some code of my own that calculated pot equity given the (simulated) opponents' holdings. This did a lot to reinforce basic stuff like wanting top pair on the flop, enthusiastically pumping flush draws, etc.

Cheers,
--JTR.

[gaming_mouse]

[ QUOTE ]
The problem with something like this is that it makes an assumption that everyone is playing the same starting hands.

[/ QUOTE ]

Yes. If you read my post again you'll see that I point this out. I made this assumption for illustrative purposes. Then I noted that you could build different (but similar) tables for different types of players and for different positions. The calculation would become more complex but still doable even in your head.

gm

[gaming_mouse]

[ QUOTE ]
A goodish sort of thing to think about.But try playing with Poker calculator or pokerStove.

[/ QUOTE ]

Problem is, products like pokerstove -- havn't tried that one specifically, so correct me if i'm wrong -- give all possible opp holdings equal weight. I'm trying to account for the fact that in real life most hands (more than half) are never played at all by typical players. This changes the odds quite a bit.

gm

[Piers]

You can only give a player a 0 or 1 chance of having each hand.

[gaming_mouse]

[ QUOTE ]
You can only give a player a 0 or 1 chance of having each hand.

[/ QUOTE ]

Given that a player has stayed in, it makes perfect sense to talk about the probability that he has a certain hand.

Of course, you are technically right that, for any particular hand XX, the player either has hand XX or doesn't. But that is an academic point.

What I meant (and it should be obvious) is this: If we were in the same situation (same flop, action, etc) with the same player many times, what percent of the time does he hold a hand with card X in it?

gm

[Piers]

The way both programs work is that you give the players ranges of hands. Each permutation in the range has equal chance taking into account the cards the other plaers have. Just download them and see for your self they are free.

poker calculator poker stove

[Steve Chase]

Hi, Mouse. How did you get 17% of holding a Q?