I'm still not 100% clear on what you want. I think I'm confused by the ROI. If I'm guessing correctly, then you want to solve for the ROI, and the ROI is equivalent to a statement about Hero's %-age chance to win the table.

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We need to know the probability distribution of results. (We have that)
We need to calculate the mean value and StDev per trial. (We can do that with the prob dist)
We need to know the number of trials. (We can stipulate this as an independent variable)
We need to choose the CI. (We can stipulate this as an independent variable)

In order to solve for the desired mean value which we equate to a win-equity, which we equate to ROI.

The lower boundary of the interval for N trials will be
N * ( EV - Z_value * StDev / SQRT(N) )
or
N * EV - SQRT(N) * Z_value * StDev

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The EV is the dot product of the equity of results and the value of results.
The EV is the mean value, so you have the mean value, and you need the StDev.
You know how to calculate StDev for a given prob dist, so you have that, too.

Next, you state a CI, and find the associated Z_value. E.g. for a C.I. of 95%, the Z_value is 1.96
Now you have to consider that this gives the middle 95%, and we don't want that. We want the top 95%.
We're not worried about the times when Hero's final value is well above the mean, only when it is well below the mean.
If we use 1.96 and only look at the lower tail, then we're really calculating the 97.5% CI.
So we actually want the Z_value associated with a CI of 90%. This middle 90% chops off 5% from the top and 5% from the bottom.
We only care about the bottom %.

Once you have the appropriate Z_value, you can plug and chug with your ROI values to get the output to be 0.
Or you can write out the math and just solve it right out, so that you put in the CI and it tells you the minimum ROI. This could be messy, since changing the ROI will change the variance, which will change the StDev, so you could end up with an ugly equation to solve.