We need to account for the build time of the worker, as well as the time required for it to mine 50 minerals.
Here is the equation
According to BWAPI, SCVs take 300 frames to make:
Workers in BW gather at ~1 mineral/second, so as on fastest there are 24 frames a second:
The cost of a worker is 50 minerals
Finally we get
So from this math it takes 1550 frames, or about 65 seconds on fastest for a worker to be built and then make back its cost. In a build order that is trying to reach a certain amount of minerals the fastest, it would be optimal to stop building workers about a minute before completion.
The following are simulation results for time to gather 1000 minerals starting with 1 worker, and controlling the number of workers built. The fastest time was with 10 workers built, 11 in total.
INFO: pred: 22223, cur: 0 work_time:22223 simulated 0 worker time: 22223 INFO: pred: 12373, cur: 1412 work_time:10961 simulated 1 worker time: 12373 INFO: pred: 9325, cur: 2118 work_time:7207 simulated 2 worker time: 9325 INFO: pred: 7919, cur: 2588 work_time:5331 simulated 3 worker time: 7919 INFO: pred: 7146, cur: 2941 work_time:4205 simulated 4 worker time: 7146 INFO: pred: 6680, cur: 3241 work_time:3439 simulated 5 worker time: 6680 INFO: pred: 6390, cur: 3541 work_time:2849 simulated 6 worker time: 6390 INFO: pred: 6211, cur: 3841 work_time:2370 simulated 7 worker time: 6211 INFO: pred: 6104, cur: 4141 work_time:1963 simulated 8 worker time: 6104 INFO: pred: 6049, cur: 4441 work_time:1608 simulated 9 worker time: 6049 INFO: pred: 6031, cur: 4741 work_time:1290 simulated 10 worker time: 6031
The final work time of 1290 frames, adding the 300 for train time, yields 1590, just above the cutoff for breakeven time to build a worker. Building another worker after that time resulted in a longer time to reach the mineral goal of 1000.
