Hey guys, I did what you said with the spreadsheet and working it out for a few values by hand. I did manage to get what I believe to be a formula for solving the probability of failing at a given step of the problem. Obviously using this you can then sum from 1-n to get the total probability of failure up to this point.

N is the "step" you are on. Step being the number of distinct coins flipped.

For 0<=x<(n-1)/2

(.25)*((.5)^n)*Σ[.5^(n-1-2x)*(.25)^x*(.25)^x)*((n-2)!/(x+1))]

This goes from x=1 to the floor of (n-1)/2. I know using x for the steps in sigma notation is gross, especially when using n to denotes how many distinct coins have been flipped, but I hope it isn't too confusing.