Sample Size: Testing a second methodology on the same population

Need some guidance ...

I have a population of a transactions whose outcomes can only complete or fail. (Binary outcome only)

Methodology #1 produces 15% completions and we have a very large number of sample outcomes.

A new methodology is proposed that may or not be better than the first. Trying the second is also not low in cost and any testing needs to be terminated ASAP if it is not superior to the first.

Here is my question:

How many new samples of Methodology #2 (on the same population) must be tested to determine if the means of each Methodologies for completions are statistically different at the 95% level?

All comments and suggestions are welcome! Thank you!

Best regards,