Probability of Recruit

python simulation:
Simulation Count: 100000000
Item                 Total Count     Expected Value
common               275763077.0     2.7576
rare                 165462732.0     1.6546
epic                 110307127.0     1.1031
legendary            82735537.0      0.8274
mythic               1653403.0       0.0165
200gold              55151834.0      0.5515
400gold              22064086.0      0.2206
600gold              9924646.0       0.0992
1stone               11029598.0      0.1103
2stone               3309378.0       0.0331
10diamond            33084016.0      0.3308
20diamond            11027583.0      0.1103
30diamond            9925446.0       0.0993
stone_per_recruit: 0.6725
stone_per_game: 2.1520

Mathematica mathematical reasoning:
stone_per_recruit: 0.672828 
stone_per_game: 2.15305 

https://preview.redd.it/ardz27ygq6ae1.png?width=1230&format=png&auto=webp&s=01545ac2a6c133fef96c62e44bf163c24afaad2e