SoB, WW and AB
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Even if the hands themselves are close to independent, as in games without community cards, your chance of beating other players will almost always be highly dependent.
In any actual situation, I would expect your probability of beating everyone to be different (usually higher) than your calculation.
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And WW wrote "However, hold'em hands, which all share the same 5-card board, tend to run closer together. As a result, the independence assumption does not apply"
I'm not quite getting what you all mean by the "independance assumption." Obviously if there is a 3-flush on board, then the hand ranges I'm assigning them have to change, and therefore my "change to beat individually" also changes. I'm not associating that idea with interdependence correctly, I believe??
Are you referring to hands counterfeiting one another with shared cards?
I was intending to use this in two ways:
1) as a rough estimate of the chances of success, if I can successfully determine a range of hands each opponent would hold (that's a risky assumption on my part! :P )
If I ignore betting tendencies and the like, knowing that I'm cutting out 70% of the real world influences that would be in effect, why can't the simple percentages be used?
I thought a rough percentage would be fairly accurate, but if the 13% vs. 30% for Aces is correct, then I may be assuming too much.
BTW- In my mind, I was thinking something like holding a big pair such as AA against loose players and doing a quick calculation of my chances against them (my original examples).
2) as a teaching tool for some players I'm working with; the lessons to be taught are ones on facing multiple opponents and the effects they have, as a group, on your hand(s).
Also, is there a limit on how I can use these calculations in a rough manner? For example, if I think I'm behind everyone and give myself a 30% chance against four others individually, my chance to beat everyone CAN'T be less than 1%, can it?
Is there some rule-of-thumb similar to counting outs on the flop, turn and river, where your addition can't make your hand more than around a 50% chance of winning? At least, that's how I understand the out-counting rule-of-thumb.
Thanks again for the help.
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