Ideal level of confidence/certainty

Staging the scenario
A casino offers you a gamble with a 1% chance of winning a try.

How many tries will it take to win at least once? The answer involves two variables, the chance of success each try and an allowance to be wrong in exchange for predictive accuracy. For this example, I choose 95% confidence, a willingness to be wrong once in twenty:

tries = log(chanceToBeWrong) / log(chanceFailureEachTry) = log(1/20) / log(99%) ≅ 300 tries

The question posed
But surely there must be a better way than to just arbitrarily select a level of confidence. It looks to me like an optimization problem, where tries are an expense to be minimized, and confidence spans across a range of 0..1

My first stab at it is to just rearrange the variables to solve for 1-confidence:

chanceToBeWrong = chanceFailureEachTry ^ numTries

But this is a surface of answers, like there's still some missing threshold, some cost/benefit crossover point(plane?)
Can anyone shed some light on how I might calculate the most appropriate level of confidence?
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Ambassador to the humans
If you're looking to do a cost/benefit analysis then at the very least you would need to know how much you would win if you do in fact win and how much it cost to play. You haven't indicated knowing either of those so it seems a bit undefined at the moment.