There are a lot of different numbers out there on the probability of successfully 6-linking an item with an Orb of fusing(and therefore the expected number of tries) in the game Path Of Exile.
So to perform an experiment, how many attempts would have to be made to be 99% confident of the probability, assuming attempts are independent of one another and that the actual probability is between [1/2000, 1/500]?
Let's say that I use 50 000 fusings and get 72 six linked items, how accurate would it be to say that the probability is 1 in 700? Should I increase the amount of fusings to be more confident??
There is no way to assign confidence to a single number when the true quantity is only known to lie in some continuous range. Instead, generally you have the liberty to set your confidence level or or your interval width but not both. Sometimes you also have the liberty to change your sample size, which can allow you to get a small interval and a high confidence level simultaneously. But generally, once you are given a sample, you can convert an interval width to a probability or vice versa. You seem to want to convert a probability (99% confidence) to an interval width.
For a large number of independent "true/false" samples like this, you would usually approximate the number of successes by a normal distribution with unknown mean and a standard deviation given by $S=\sqrt{k(1-k/n))}$ where $k$ is the number of success you observed in $n$ samples. Then the 99% confidence interval for the success probability is approximately $((k-2.575S)/n,(k+2.575S)/n)$, where the 2.575 came from the normal distribution and the confidence level you picked.
In your example, this would be $0.0014 \pm 0.0004$ (where I rounded such that only one uncertain digit is retained), so somewhere between 1 in 1000 and 1 in 550. More samples would help, but as usual it is difficult to estimate success probabilities with good relative error when the true value is close to zero.