Cumulative Confidence Interval
Hello. I'm trying to determine how many attempts I need to gather resources for in a game. Each attempt costs the same number and type of resources, but the odds of a success go down as you go on. Only one success per attempt is possible. Each chance is separate, not attempted for simultaneously. The specific calculation that brought about this question has the following variables.
26 successes at 10% chance of success per attempt
26 successes at 7% chance of success per attempt
21 successes at 5% chance of success per attempt
I'm quite able to use overunder with binomial distribution to figure a confidence level any of those variables out individually. Cumulatively, however, while I could hazard a guess, I am far less confident on. I'm not skilled enough in programming to throw a sim at the problem, and I've come up empty in searching for extant formulae for such a question, though highly suspect there would be one.
The minimum goal is to calculate the number of attempts needed to achieve these 73 successes at 95% confidence.
The highest goal would be a formula or easily accessible calculator with the ability to plug in any number of variables as shown above.
I would prefer answers be written with any variables labeled clearly, as my memory of common mathematical shorthand is rusty (for example, I do not recall which units m or c are written in in e=mc^2, and only recall the e is in Joules due to a Robot Chicken sketch).
Thank you.

“the minimum number of attempts needed to achieve these 73 successes at 95% confidence.” Sorry, but this is completely meaningless. The minimum number of attempts is 73.

A problem with my phrasing perhaps, as you're certainly not 95% confident to get all 73 successes after 73 attempts with probabilities like that. If you'd have a better way to put it, I'd like to know.

I think I understand what you are asking for: you have 3 different processes, each with different success chances that you need to get a certain number of successes on. You want to put an interval on the total number of trials necessary to complete all the successes for all the trials. Unfortunately, it is very unlikely that the solution to this problem has a nice formula. The best I believe you can do is to write some code to calculate the answer. You do not need to simulate it, however.

This is to say, Cocoli, I could provide an answer to this question, but it would have to come in form of some code, rather than a clean formula, because such a formula does not exist.

" the odds of a success go down as you go on" You mean they go down as you make an attempt or they go down as you get a success?

I think you want a confidence interval for the expected number of attempts to get 73 successes. Am I correct?

The chances keep going down after this? "21 successes at 5% chance of success per attempt" What is the formula that relates number of successes with probabilities of success?
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