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 over-under 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).
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