Back to the original question
Well, this would depend on your definition of “usually”. We can break this down statistically for the odds of pulling any gold servant, period, by adding up the odds for every possible scenario:
(odds of pulling 4* and no 5*) + (odds of pulling 5* and no 4*) + (odds of pulling both 4* and 5*)
Using the formulas we have above, that would be something like
100*((1-.97^x)(.99^x) + (1-.99^x)(.97^x) + (1-.97^x)(1-.99^x))
Now you can use an online graphing calculator like https://www.desmos.com/calculator to show you the curve (zoom out as necessary) and click on and follow the curve, with x as your # of rolls and y as your percentage to pull a gold servant.
You can see at 18 rolls (54 quartz), you have a 51.77% chance to pull any gold servant, at 33 rolls (99 quartz) you have a 73.73% chance. You don’t hit a 90% chance until 57 rolls (171 quartz). You could theoretically roll 114 times (342 quartz) and be in that unlucky 1% which still fails to hit a gold servant.
Probability curve should look something like this:
Mathematicians are free to correct me, my statistics knowledge is pretty rusty.