Thanks for posting it. Would you be willing to share the formulas?
I would if I had one.
I didn't take the time to derive a closed form solution, I simply "brute-forced" it with an Excel spreadsheet. The solution is actually a collection of pages (15 tabs in a spreadsheet) some of them containing over 30,000 lines of data. Not that easy to share, unfortunately...
Basically the spreadsheet replicates every possible outcome for each scenario (5 shot, 6 shot, etc.), then calculates the probability of each outcome and sums the probabilities of each outcome that meets the success criterion.
Not pretty or elegant, but it works and provides some easy ways to double check to make sure the results make sense.