Sorry it took so long to reply to you guys. I greatly appreciate your efforts, but I didn't want to reply until I'd done some work on this myself.
I just did the rigorous long form probability distribution for 100 trials and everything is in confirmation of jackvance's numbers. However, in doing so I discovered that what I really want is the likelihood of having a positive outcome after a stated number of trials, not necessarily an outcome above the mean. For a zero EV play, clearly these are the same thing but for a positive expected value play they are not.
Also, I tink it really it isn't jackvance's value that I want, it is the cumulative probability of all positive outcomes, not a cumulative probability of mean-proximity-based outcomes.
MMM, I think you're misunderstanding my meaning a little. Of course I believe that if our equity is > 1/3, that calling is +EV no matter what. My point is that making calls for big bets that are very thin is practically -EV. These calls are a poor use of the money it takes to make them. My hypothesis is that if you state an event frequency (say n trials in a six month period, for example) and a bankroll amount, there will be a hard number for how much equity you need in excess of breakeven in order to justify calling. As EV increases from zero, for a significant number of trials, the expectation that we will at least breakeven should approach 100%. For example, in this pot size river bet, I'm curious to know the EV required to have a 95%, 90%, or 75% chance of a positive result after 100 trials. As it turns out:
>95% chance of +$ ----- 41.6% equity ----- 0.248 units +EV/trial
>90% chance of +$ ----- 39.8% equity ----- 0.193 units +EV/trial
>75% chance of +$ ----- 36.8% equity ----- 0.104 units +EV/trial
>50% chance of +$ ----- 33.6% equity ----- 0.008 units +EV/trial
I for one find it fairly surprising that you need over 40% equity to have a 95% chance of being up after 100 calls. Clearly, an attempt to reduce one's variance by so much would come with a pretty huge sacrifice of EV. To even get a 75% chance, you have to sacrifice 100*0.104 = over 10 pot size bets worth of EV. That said, I think under minor bankroll constraints there's still a reasonable case for folding hands in the 33-35% range to save variance.
The google sheet i made for this is pretty sweet, if either of you want a link to it for your own curiosity, PM me.
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