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