Probability question (math nerds welcome)

[Renton]

Yeah that's certainly what I'm looking for, assuming it is correct. Now I want to figure out how to quickly do this for a variety of % ranges i.e. 99 90 75 etc, and manipulate the population a little and see what results. For example if we change our equity on the river from 33% to 37%, how much different do those numbers look? I'm gonna set up a google sheet for this.

My end goal is to test a hypothesis that I have which is the following: Very thin plays for big bets aren't just marginally +EV, they're -EV. Practically, at the sample sizes humans are capable of experiencing, particularly during reasonable spans of time (say around 6 to 24 months of play), if you can't be reasonably confident that the play will result in equity > the mean, and your bankroll and sanity aren't nearly infinite, you should just pass on these spots. So I want to know "how thin is okay" for a variety of common poker situations, and eventually develop an intuitive sense of this.

[MadMojoMonkey]

Yeah that's certainly what I'm looking for, assuming it is correct.

I agree with JV's numbers.

Now I want to figure out how to quickly do this for a variety of % ranges i.e. 99 90 75 etc, and manipulate the population a little and see what results.

This is gonna be an easy task.
The multiplicative factor of 1.92 is the only thing that changes when you're looking at different CI %-ages. For simplicity, I'm going to use the "infinite trials" value, which should only have a slight error in the 3rd sig.fig.
75% CI -> 1.15
90% CI -> 1.64
95% CI -> 1.96
99% CI -> 2.58

On Excel, you can get this number by putting the %-age in cell A1. Into another cell, enter
=NORMSINV(1-(1-A1)/2)

For example if we change our equity on the river from 33% to 37%, how much different do those numbers look? I'm gonna set up a google sheet for this.

Well, first off, if your equity goes from 33% to 37% and the bet sizes remain the same, then the EV will be positive. We know that 33% equity is exactly break-even with those bet amounts. So greater equity means greater EV, and EV was 0, so def. +EV.

My end goal is to test a hypothesis that I have which is the following: Very thin plays for big bets aren't just marginally +EV, they're -EV.

No matter how wide the variance, if a bet is +EV, then the expectation is to win more than is lost after many bets.

Practically, at the sample sizes humans are capable of experiencing, particularly during reasonable spans of time (say around 6 to 24 months of play), if you can't be reasonably confident that the play will result in equity > the mean, and your bankroll and sanity aren't nearly infinite, you should just pass on these spots. So I want to know "how thin is okay" for a variety of common poker situations, and eventually develop an intuitive sense of this.

We know that all +EV bets are good in the long term, but sanity has time constraints. I feel like the obv answer is if you want to take poker seriously, you have to embrace the variance, not avoid it. The assumed goal is to seek out the max EV in any situation, even if a lower EV play has lower variance.

I'm interested in your conclusions.

[MadMojoMonkey]

Can you address what the results of my long form calculation at the end of the post would be? Would it be the same as the 95% confidence interval or something different entirely. If the latter, then confidence interval is definitely not the value that interests me.

In cell B1 input 0.33
In cell C1 input =(1-B1)
{I format these as percents, but whatever}

In cell A2 input 0
In cell A3 input 1
In cell A4 input 2
{select cells A2 - A4, drag the bottom-right corner of the selected group down so that column A has numbers 0 - 100}

In cell B2 input =COMBIN(100,A2) * B$1^A2 * C$1^(100-A2)
{double-click the bottom right corner of B2 after entering the above formula to fill column B in just the right way.}

In cell C2 input =B2
In cell C3 input =SUM(B$2:B3)
{double-click the bottom right corner of C3 after entering the above formula to fill column C in just the right way.}

Make graphs of Columns B and C, with Column A as the x-axis.



The top is the probability distribution for X wins after 100 trials.
The bottom is the cumulative distribution for X wins after 100 trials. (It is the integral of the top function.)
Bottom may be the more informative graph.

A cursory glance pulls out the 90% CI as [25, 40].
I excluded the lower and upper 5%, leaving the middle 90%.